Learning and Intelligent Optimization For Material Design Innovation

Roberto Battiti
Dmitri E. Kvasov
Yaroslav D. Sergeyev (Eds.)
LNCS 10556
Learning and
Intelligent Optimization
11th International Conference, LION 11
Nizhny Novgorod, Russia, June 1921, 2017
Revised Selected Papers
123
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Roberto Battiti Dmitri E. Kvasov

Yaroslav D. Sergeyev (Eds.)
Learning and
Intelligent Optimization
11th International Conference, LION 11
Nizhny Novgorod, Russia, June 1921, 2017
Revised Selected Papers
123
Editors
Roberto Battiti Yaroslav D. Sergeyev
University of Trento University of Calabria
Trento Rende
Italy Italy
and and
Lobachevsky University of Nizhny Lobachevsky University of Nizhny

Novgorod Novgorod
Nizhny Novgorod Nizhny Novgorod
Russia Russia
Dmitri E. Kvasov
University of Calabria
Rende
Italy
and
Lobachevsky University of Nizhny

Novgorod
Nizhny Novgorod
Russia
ISSN 0302-9743 ISSN 1611-3349 (electronic)

Lecture Notes in Computer Science
ISBN 978-3-319-69403-0 ISBN 978-3-319-69404-7 (eBook)
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Preface
This volume edited by R. Battiti, D.E. Kvasov, and Y.D. Sergeyev contains peer-
reviewed papers from the 11th Learning and Intelligent International Optimization
conference (LION-11) held in Nizhny Novgorod, Russia, during June 1921, 2017.
The LION-11 conference has continued the successful series of the constantly
expanding and worldwide recognized LION events (LION-1: Andalo, Italy, 2007;
LION-2 and LION-3: Trento, Italy, 2008 and 2009; LION-4: Venice, Italy, 2010;
LION-5: Rome, Italy, 2011; LION-6: Paris, France, 2012; LION-7: Catania, Italy,
2013; LION-8: Gainesville, USA, 2014; LION-9: Lille, France, 2015; LION-10:
Ischia, Italy, 2016). This edition was organized by the Lobachevsky University of
Nizhny Novgorod, Russia, as one of the key events of the Russian Science Foundation
project No. 15-11-30022 Global Optimization, Supercomputing Computations, and
Applications. Like its predecessors, the LION-11 international meeting explored
advanced research developments in such interconnected elds as mathematical pro-
gramming, global optimization, machine learning, and articial intelligence. Russia has
a long tradition in optimization theory, computational mathematics, and intelligent
learning techniques (in particular, cybernetics and statistics), therefore, the location of
LION-11 in Nizhny Novgorod was an excellent occasion to meet researchers and
consolidate research and personal links.
More than 60 participants from 15 countries (Austria, Belgium, France, Germany,
Hungary, Italy, Lithuania, Portugal, Russia, Serbia, Switzerland, Taiwan, Turkey, UK,
and USA) took part in the LION-11 conference. Four plenary lecturers shared their
current research directions with the LION-11 participants:
Renato De Leone, Camerino, Italy: The Use of Grossone in Optimization: A
Survey and Some Recent Results
Nenad Mladenovic, Belgrade, Serbia: Less Is More Approach in Heuristic
Optimization
Panos Pardalos, Gainesville, USA: Quantication of Network Dissimilarities and
Its Practical Implications
Julius ilinskas, Vilnius, Lithuania: Deterministic Algorithms for Black
Box Global Optimization
Moreover, three tutorials were also presented during the conference:
Adil Erzin, Novosibirsk, Russia: Some Optimization Problems in the Wireless
Sensor Networks
Mario Guarracino, Naples, Italy: Laplacian-Based Semi-supervised Learning
Yaroslav Sergeyev, University of Calabria, Italy, and Lobachevsky University of
Nizhny Novgorod, Russia: Numerical Computations with Innities and
Innitesimals
VI Preface
A total of 20 long papers and 15 short papers were accepted for publication in this
LNCS volume after thorough peer reviewing (up to three review rounds for some
manuscripts) by the members of the LION-11 Program Committee and independent
reviewers. These papers describe advanced ideas, technologies, methods, and appli-
cations in optimization and machine learning. This volume also contains the paper
of the winner (Francesco Romito, Rome, Italy) of the second edition of the
Generalization-Based Contest in Global Optimization (GENOPT: http://genopt.org).
The editors thank all the participants for their dedication to the success of LION-11
and are grateful to the reviewers for their valuable work. The support of the
Springer LNCS editorial staff is greatly appreciated.
The editors express their gratitude to the organizers and sponsors of the LION-11
international conference: Lobachevsky University of Nizhny Novgorod, Russia;
Russian Science Foundation; EnginSoft Company, Italy; NTP Truboprovod, Russia;
and the International Society of Global Optimization. Their support was essential for
the success of this event.
August 2017 Roberto Battiti

Dmitri E. Kvasov
Yaroslav D. Sergeyev
Organization
General Chair
Yaroslav Sergeyev University of Calabria, Italy and Lobachevsky
University of Nizhny Novgorod, Russia
Steering Committee
Roberto Battiti (Head) University of Trento, Italy and Lobachevsky
Holger Hoos University of British Columbia, Canada
Youssef Hamadi cole Polytechnique, France
Mauro Brunato University of Trento, Italy
Thomas Sttzle Universit Libre de Bruxelles, Belgium
Christian Blum Spanish National Research Council
Martin Golumbic University of Haifa, Israel
Marc Schoenauer Inria Saclay, le-de-France
Xin Yao University of Birmingham, UK
Benjamin Wah The Chinese University of Hong Kong
and University of Illinois, USA
Panos Pardalos University of Florida, USA
Technical Program Committee

Annabella Astorino ICAR-CNR, Italy
Roberto Battiti University of Trento, Italy and Lobachevsky
Bernd Bischl Ludwig Maximilians University Munich, Germany
Christian Blum Spanish National Research Council
Mauro Brunato University of Trento, Italy
Sonia Caeri cole Nationale de lAviation Civile, France
Andre de Carvalho University of So Paulo, Brazil
John Chinneck Carleton University, Canada
Andre Cire University of Toronto Scarborough, Canada
Renato De Leone University of Camerino, Italy
Luca Di Gaspero University of Udine, Italy
Bistra Dilkina Georgia Institute of Technology, USA
Adil Erzin Sobolev Institute of Mathematics SB RAS, Russia
Giovanni Fasano University CaFoscari of Venice, Italy
Paola Festa University of Naples Federico II, Italy
VIII Organization
Antonio Fuduli University of Calabria, Italy

David Gao Federation University, Australia
Martin Golumbic University of Haifa, Israel
Vladimir Grishagin Lobachevsky University of Nizhny Novgorod, Russia
Tias Guns Vrije Universiteit Brussel, Belgium
Youssef Hamadi cole Polytechnique, France
Frank Hutter Albert-Ludwigs-Universitt Freiburg, Germany
George Katsirelos MIAT-INRA, France
Michael Khachay Krasovsky Institute of Mathematics and Mechanics UB
RAS, Russia
Oleg Khamisov Melentiev Institute of Energy Systems SB RAS, Russia
Yury Kochetov Sobolev Institute of Mathematics SB RAS, Russia
Lars Kotthoff University of British Columbia, Canada
Dmitri Kvasov University of Calabria, Italy and Lobachevsky University
of Nizhny Novgorod, Russia
Dario Landa-Silva The University of Nottingham, UK
Hoai An Le Thi Universit de Lorraine, France
Daniela Lera University of Cagliari, Italy
Vasily Malozemov Saint Petersburg State University, Russia
Marie-Elonore CRISTAL/Inria/Lille University, France
Marmion
Silvano Martello University of Bologna, Italy
Kaisa Miettinen University of Jyvskyl, Finland
Nenad Mladenovic Mathematical Institute SANU, Serbia
Evgeni Nurminski Far Eastern Federal University, Russia
Barry OSullivan University College Cork, Ireland
Panos Pardalos University of Florida, USA
Remigijus Paulaviius Imperial College London, UK and Vilnius University,
Lithuania
Thomas Pock Graz University of Technology, Austria
Mikhail Posypkin Dorodnicyn Computing Centre, FRC CSC RAS, Russia
Oleg Prokopyev University of Pittsburgh, USA
Helena Ramalhinho Universitat Pompeu Fabra, Spain
Loureno
Massimo Roma Sapienza University of Rome, Italy
Francesca Rossi University of Padova, Italy and Harvard University, USA
Valeria Ruggiero University of Ferrara, Italy
Horst Samulowitz IBM T.J. Watson Research Center, USA
Marc Schoenauer Inria Saclay, le-de-France
Meinolf Sellmann General Electric Global Research, USA
(Chair) University of Nizhny Novgorod, Russia
Carlos Soares University of Porto, Portugal
Alexander Strekalovskiy Matrosov Institute for System Dynamics and Control
Theory SB RAS, Russia
Thomas Sttzle Universit Libre de Bruxelles, Belgium
Organization IX
ric Taillard University of Applied Science of Western Switzerland

Tatiana Tchemisova University of Aveiro, Portugal
Gerardo Toraldo University of Naples Federico II, Italy
Michael Trick Carnegie Mellon University, USA
Daniele Vigo University of Bologna, Italy
Petr Vilm IBM Czech, Czech Republic
Luca Zanni University of Modena and Reggio Emilia, Italy
Anatoly Zhigljavsky Cardiff University, UK
Antanas ilinskas Vilnius University, Lithuania
Julius ilinskas Vilnius University, Lithuania
Additional Reviewers
Steven Adriaensen Marat Mukhametzhanov

Yair Censor Duy Nhat Phan
Elena Chistyakova Soumyendu Raha
Ivan Davydov Ivan Takhonov
Alessandra De Rossi Ider Tseveendorj
Giuseppe Fedele Vo Xuanthanh
Jonathan Gillard
Local Organization Committee

Roman Strongin (Chair) Lobachevsky University of Nizhny Novgorod, Russia
Victor Kasantsev Lobachevsky University of Nizhny Novgorod, Russia
(Vice-chair)
Vadim Saygin Lobachevsky University of Nizhny Novgorod, Russia
(Vice-chair)
Konstantin Barkalov Lobachevsky University of Nizhny Novgorod, Russia
Lev Afraimovich Lobachevsky University of Nizhny Novgorod, Russia
Dmitri Balandin Lobachevsky University of Nizhny Novgorod, Russia
Victor Gergel Lobachevsky University of Nizhny Novgorod, Russia
Vladimir Grishagin Lobachevsky University of Nizhny Novgorod, Russia
Dmitri Kvasov Lobachevsky University of Nizhny Novgorod, Russia
Iosif Meerov Lobachevsky University of Nizhny Novgorod, Russia
Grigory Osipov Lobachevsky University of Nizhny Novgorod, Russia
Mihail Prilutskii Lobachevsky University of Nizhny Novgorod, Russia
Alexadner Sysoyev Lobachevsky University of Nizhny Novgorod, Russia
Dmitri Shaposhnikov Lobachevsky University of Nizhny Novgorod, Russia
Ekaterina Goldinova Lobachevsky University of Nizhny Novgorod, Russia
X Organization
Acknowledgements. The organization of the LION-11 conference was supported by

the Russian Science Foundation, project No. 15-11-30022 Global optimization,
supercomputing computations, and applications. The organizers also gratefully
acknowledge the support of the following sponsors: EnginSoft company, Italy; NTP
Truboprovod, Russia; and the International Society of Global Optimization.
Contents
Long Papers
An Importance Sampling Approach to the Estimation

of Algorithm Performance in Automated Algorithm Design . . . . . . . . . . . . . 3
Steven Adriaensen, Filip Moons, and Ann Now
Test Problems for Parallel Algorithms of Constrained Global Optimization. . . 18

Konstantin Barkalov and Roman Strongin
Automatic Configuration of Kernel-Based Clustering:

An Optimization Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Antonio Candelieri, Ilaria Giordani, and Francesco Archetti
Solution of the Convergecast Scheduling Problem on a Square

Unit Grid When the Transmission Range is 2 . . . . . . . . . . . . . . . . . . . . . . . 50
Adil Erzin
A GRASP for the Minimum Cost SAT Problem . . . . . . . . . . . . . . . . . . . . . 64

Giovanni Felici, Daniele Ferone, Paola Festa, Antonio Napoletano,
and Tommaso Pastore
A New Local Search for the p-Center Problem Based on the Critical
Vertex Concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Daniele Ferone, Paola Festa, Antonio Napoletano,
and Mauricio G.C. Resende
An Iterated Local Search Framework with Adaptive Operator

Selection for Nurse Rostering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Angeliki Gretsista and Edmund K. Burke
Learning a Reactive Restart Strategy to Improve Stochastic Search . . . . . . . . 109

Serdar Kadioglu, Meinolf Sellmann, and Markus Wagner
Efficient Adaptive Implementation of the Serial Schedule Generation

Scheme Using Preprocessing and Bloom Filters . . . . . . . . . . . . . . . . . . . . . 124
Daniel Karapetyan and Alexei Vernitski
Interior Point and Newton Methods in Solving High Dimensional

Flow Distribution Problems for Pipe Networks . . . . . . . . . . . . . . . . . . . . . . 139
Oleg O. Khamisov and Valery A. Stennikov
XII Contents
Hierarchical Clustering and Multilevel Refinement for the Bike-Sharing

Station Planning Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Christian Kloimllner and Gnther R. Raidl
Decomposition Descent Method for Limit Optimization Problems . . . . . . . . . 166

Igor Konnov
RAMBO: Resource-Aware Model-Based Optimization with Scheduling

for Heterogeneous Runtimes and a Comparison with Asynchronous
Model-Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Helena Kotthaus, Jakob Richter, Andreas Lang, Janek Thomas,
Bernd Bischl, Peter Marwedel, Jrg Rahnenfhrer, and Michel Lang
A New Constructive Heuristic for the No-Wait Flowshop

Scheduling Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Lucien Mousin, Marie-Elonore Kessaci, and Clarisse Dhaenens
Sharp Penalty Mappings for Variational Inequality Problems . . . . . . . . . . . . 210

Evgeni Nurminski
A Nonconvex Optimization Approach to Quadratic Bilevel Problems . . . . . . 222

Andrei Orlov
An Experimental Study of Adaptive Capping in irace . . . . . . . . . . . . . . . . . 235

Leslie Prez Cceres, Manuel Lpez-Ibez, Holger Hoos,
and Thomas Sttzle
Duality Gap Analysis of Weak Relaxed Greedy Algorithms . . . . . . . . . . . . . 251

Sergei P. Sidorov and Sergei V. Mironov
Controlling Some Statistical Properties of Business Rules Programs . . . . . . . 263

Olivier Wang and Leo Liberti
GENOPT Paper
Hybridization and Discretization Techniques to Speed Up Genetic

Algorithm and Solve GENOPT Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 279
Francesco Romito
Short Papers
Identification of Discontinuous Thermal Conductivity Coefficient

Using Fast Automatic Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Alla F. Albu, Yury G. Evtushenko, and Vladimir I. Zubov
Comparing Two Approaches for Solving Constrained Global

Optimization Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Konstantin Barkalov and Ilya Lebedev
Contents XIII
Towards a Universal Modeller of Chaotic Systems . . . . . . . . . . . . . . . . . . . 307

Erik Berglund
An Approach for Generating Test Problems of Constrained

Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
Victor Gergel
Global Optimization Using Numerical Approximations of Derivatives . . . . . . 320

Victor Gergel and Alexey Goryachih
Global Optimization Challenges in Structured Low Rank Approximation . . . . 326

Jonathan Gillard and Anatoly Zhigljavsky
A D.C. Programming Approach to Fractional Problems . . . . . . . . . . . . . . . . 331

Tatiana Gruzdeva and Alexander Strekalovsky
Objective Function Decomposition in Global Optimization . . . . . . . . . . . . . . 338

Oleg V. Khamisov
Projection Approach Versus Gradient Descent for Networks Flows

Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Alexander Yu. Krylatov and Anastasiya P. Shirokolobova
An Approximation Algorithm for Preemptive Speed Scaling Scheduling

of Parallel Jobs with Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Alexander Kononov and Yulia Kovalenko
Learning and Intelligent Optimization for Material Design Innovation . . . . . . 358

Amir Mosavi and Timon Rabczuk
Statistical Estimation in Global Random Search Algorithms

in Case of Large Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Andrey Pepelyshev, Vladimir Kornikov, and Anatoly Zhigljavsky
A Model of FPGA Massively Parallel Calculations

for Hard Problem of Scheduling in Transportation Systems . . . . . . . . . . . . . 370
Mikhail Reznikov and Yuri Fedosenko
Accelerating Gradient Descent with Projective Response

Surface Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
Alexander Senov
Emmental-Type GKLS-Based Multiextremal Smooth Test Problems

with Non-linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Ya.D. Sergeyev, D.E. Kvasov, and M.S. Mukhametzhanov
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Long Papers
An Importance Sampling Approach
to the Estimation of Algorithm Performance
in Automated Algorithm Design
Steven Adriaensen(B) , Filip Moons, and Ann Nowe
Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

steven.adriaensen@vub.be
Abstract. In this paper we consider the problem of estimating the rel-

ative performance of a given set of related algorithms. The predominant,
general approach of doing so involves executing each algorithm instance
multiple times, and computing independent estimates based on the per-
formance observations made for each of them. A single execution might
be expensive, making this a time-consuming process. We show how an
algorithm in general can be viewed as a distribution over executions;
and its performance as the expectation of some measure of desirability
of an execution, over this distribution. Subsequently, we describe how
Importance Sampling can be used to generalize performance observa-
tions across algorithms with partially overlapping distributions, amor-
tizing the cost of obtaining them. Finally, we implement the proposed
approach as a Proof of Concept and validate it experimentally.
Keywords: Performance evaluation Algorithms Importance Sam-

pling Programming by Optimization Automated algorithm synthesis
1 Introduction
Often, there are many ways to solve a given problem. However, not all of these
are equally good. In the Algorithm Design Problem [2,14] (ADP) we are to
nd the best way to solve a given problem; e.g. using the least computational
resources (time, memory etc.), and/or maximizing the quality of the solutions
obtained. In a sense it is the problem of how to best solve a given problem.
To date, algorithms for many real-world problems are most commonly
designed following a manual, ad-hoc, trial & error approach, making algorithm
design a tedious and costly process, often leading to mediocre results. Recently,
Programming by Optimization [8] (PbO) was proposed as an alternative design
paradigm. In PbO, dicult choices are deliberately left open at design time, thus
programming a family of algorithms (design space), rather than a single algo-
rithm. Subsequently, optimization methods are applied to automatically deter-
mine the best algorithm instance (design) for a specic use-case. Often, the latter
Steven Adriaensen is funded by a Ph.D. grant of the Research Foundation Flanders
(FWO).
c Springer International Publishing AG 2017
R. Battiti et al. (Eds.): LION 2017, LNCS 10556, pp. 317, 2017.
https://doi.org/10.1007/978-3-319-69404-7_1
4 S. Adriaensen et al.
requires evaluating each design numerous times, as we are typically interested

in performance across multiple inputs and observations thereof might be noisy.
In turn, a single evaluation can be expensive as it involves algorithm execution,
followed by some sort of quantication of its desirability. As the design space
considered is often huge, this quickly becomes a time-consuming process.
In this paper we propose a novel way of estimating the relative performance of
a given set of related algorithms, targeted at reducing the number of evaluations
needed to solve a given ADP, i.e. solve it faster. This approach is based on the
observation that dierent algorithm instances in the design space might, with
some likelihood, generate the exact same execution, i.e. perform the exact same
sequence of instructions. Some motivating examples are the following:
Conditional parameters in algorithm conguration [10]: Often not all para-

meters are used during every execution of an algorithm, i.e. the likelihood of
an execution does only depend on the values of those parameters that were
actually used.
Selecting a cooling schedule for Simulated Annealing [15] (SA): A cooling
schedule, determines the likelihood of accepting worsening proposals in SA
at each point in time. Most schedules, at each iteration i, accept a worsen-
ing proposals with a probability 0 < pi < 1, i.e. can generate any possible
execution.
The crux is that even though an execution e might have been obtained using
some design c, it might as well (with some likelihood) have been generated
using a dierent design c . As such, the observed desirability of e does not only
provide information about the performance of c, but also about that of c . In
this work we describe how Importance Sampling (IS) can be used to combine all
performance observations relevant to an algorithm, into a consistent estimator
of its performance, assuming we are able to compute the (relative) likelihood of
generating a given execution, using a given design.
The remainder of this paper is structured as follows. First, in Sect. 2, we
formally dene the ADP and related concepts such as the design space, desir-
ability of an execution and algorithm performance. Subsequently, in Sect. 3.1, we
summarize contemporary approaches to performance estimation in PbO; before
introducing our IS approach in Sect. 3.2, discussing its benets in Sect. 4 and
examining its theoretical feasibility in Sect. 5. Finally, we discuss some chal-
lenges encountered when trying to implement IS in practice, and how we have
addressed these in the implementation of a Proof of Concept in Sect. 6, which
we validate experimentally in Sect. 7, before concluding in Sect. 8.
2 The Algorithm Design Problem (ADP)

Informally, we refer to the ADP as any attempt to (automatically) decide how
to best solve a given problem. In the following, we formalize this notion and
introduce concepts and notation used throughout this article.
An IS Approach to the Estimation of Algorithm Performance 5
Let C be the set of alternative algorithm instances considered, i.e. the design
space. Let D denote a distribution over X, the set of possible inputs (e.g. problem
instances to be solved and budgets available for doing so). Let E be the execu-
tion space, i.e. the set of all possible executions of any c C, on any x X.
Let P r(e|c) denote the likelihood that executing algorithm instance c, on an
input x D, results in an execution e, and f : E R a function quantifying
the desirability of an execution. We dene algorithm performance as a function
o : C R:

o(c) = P r(e|c)f (e). (1)
eE
i.e. every algorithm instance in the design space can be viewed as a distribution
over executions, whose performance corresponds to the expectation of f , over
this distribution (as illustrated in Fig. 2). Remark that executing an algorithm
instance (on x D) corresponds to sampling from its corresponding distribution.
The objective in the ADP is to nd c = arg maxcC o(c).
f(e)
Design Space (C)
Pr
o Execution Space (E)
Pr(e|c)
f
c1 c2
Performance Space ( )
Executions (E)
Fig. 1. A diagram showing how P r and Fig. 2. Distribution perspective

f relate designs to their performance.
Thus far, optimizers of choice for PbO have been algorithm congurators,
i.e. Programming by Conguration (PbC). Here, alternative designs are rep-
resented as congurations and congurators search the space of congurations C
for one maximizing performance. As such, the ADP is treated as an Algorithm
Conguration Problem [10] (ACP). Our formulation above is very similar and
can in fact be seen as a simplied, specialization of the ACP. The most relevant
dierence, in context of this paper, lies in our denition of o. As argued in [2],
PbC treats algorithm evaluation as a (stochastic) black box function C R.
Above, we dene this mapping as a consequence of algorithm execution (see
Fig. 1), i.e. our choice of design c aects execution e in a particular way (P r),
which in turn relates to its observed desirability (f ). The performance estimation
technique proposed in this paper exploits this feature, assuming P r (and f ) to
be computable (black box) functions. In Sect. 5 we argue why this assumption
is reasonable in context of PbO.
3 Performance Estimation in PbO

3.1 Prior Art
Computing o(c) exactly, e.g. using (1), is often intractable due to the size of
E. As such, general conguration frameworks rely on estimates o(c). Most (e.g.
[3,10,12]) maintain an independent sample average as performance estimate for
each design c. They obtain a set of executions Ec by repeatedly
evaluating c
(see Figs. 3 and 4) and estimate o(c) as o(c) = o(c) = |E1 | eE f (e). Here, so
c c
called Sequential Model-Based Optimization (SMBO) frameworks are a notable
exception (e.g. SMAC [9]). SMBO frameworks maintain a regression model
M : C R (a.k.a. response surface model) trained using all previous per-
formance observations. As such M gives us a performance prediction for any
design. Note that these predictions are, to the best of our knowledge, only used
to guide the search (identify unexplored, potentially interesting areas in the con-
guration space), i.e. to decide on which design to evaluate next. The choice of
incumbent, i.e. which design to return as solution at any time, is based solely on
performance observations obtained using the design itself.
f(e)
f(e)
c c
Pr(ec)
Pr(ec)
c c
Executions (E) Executions (E)
Fig. 3. Two dierent designs Fig. 4. Two similar designs (color

gure online)
3.2 An Importance Sampling Approach

The use of independent estimates is reasonable if the distributions in the design
space are disjoint, as in Fig. 3. However, in what follows, we will argue that we
can do better if two or more distributions overlap, as in Fig. 4, i.e. if there exists
an execution that can be generated by at least two dierent algorithm instances:
e E ci , cj C : P r(e|ci ) P r(e|cj ) > 0. When distributions overlap,
evaluations of one design, potentially provide information about another. As
evaluations can be extremely expensive, we would like to maximally utilize the
information they provide; e.g. in Fig. 4, also use the red crosses to estimate the
expectation over the blue distribution and vice versa.
Importance Sampling (IS) is a general technique for estimating properties

of one distribution, using samples generated by another. IS originates from the
eld of rare event sampling [6], where some events might strongly aect some
statistic, yet only occur rarely under the nominal distribution P. To reduce
the estimation error on the sample average, it is benecial to draw samples
from a dierent distribution G, called the importance or generating distribution,
which increases the likelihood of sampling these important events. To account
for having sampled from this other distribution, we have to adjust our estimate
by weighing observations by the likelihood ratio ( P G ). In this paper, we propose
a dierent use of IS, i.e. to combine all performance observations f (e) relevant
for a design c (i.e. P r(e|c) > 0) into a consistent estimator of its performance.
Concretely, let E be the sample of executions obtained thus far, with E G
(e.g. see Fig. 5), the IS estimate of the performance of any c is then given by

wc (e) f (e) P r(e|c)
o(c) = o(c) = eE with wc (e) = . (2)
eE wc (e) G(e)
where G(e) is the likelihood of generating e using G. While the estimate of each
design c is based on the same E ; the weight function wc will be dierent, weighing
performance observations according to their relevance to c; e.g. in Fig. 6, observa-
tions on the left and right hand side are more relevant for c1 and c2 respectively.
o(c) is a consistent estimate of o(c), as long as wc (e) is bounded, e E. In prac-
tice, it is also important that we can actually compute G(e) for any e. We will
discuss the choice of G in more detail in Sect. 6.1, but for now it should be clear
that both conditions are met if G is some mixture of all c C, i.e.

G(e) = (c) P r(e|c). (3)
cC
where (c) > 0 is the likelihood that c is used. Note that wc (e) (c) 1
holds.
1
Furthermore, remark that if distributions do not overlap, wc (e) = (c) and o = o.
f(e)
f(e)
c1 c2 c1 c2
Pr(e|c)
wc
Executions (E) Executions (E)
Fig. 5. G as equal mixture Fig. 6. Design specic weights

4 Envisioned Benefits
In this section we will discuss the benets of using IS (2) to estimate algorithm
performance in an ADP setting. In addition, we will illustrate some of these
experimentally for the abstract setup shown in Figs. 2, 3, 4, 5 and 6. Here, our
design space consists of 2 normal distributions c1 = N (1 , 1) and c2 = N (2 , 1).
Our objective is to determine which of these has the greatest mean, based on
samples E generated by alternately sampling each. To make this more challeng-
ing we add uniform white noise in [1, 1] to these observations. Results shown
are averages of 1000 independent runs, generating 1000 samples each. Obviously,
this particular instance is not representative for the ADP. Nonetheless, we would
like to argue that the observations in this section generalize to the full ADP set-
ting; e.g. computing P r requires us to actually know , i.e. o, this however is a
peculiarity of this simple setup and is denitely not the case in general (see also
Sect. 5). As IS treats P r as a black box, this fact did not aect the generality of
our results. For the critical reader, an analogous argument is made in [1] using
a somewhat more realistic ADP, benchmark 1 (see Sect. 7.1), as a running
example.
First, IS increases data-eciency. By using a single performance observation
in the estimation of many designs, we amortize the cost of obtaining it, and will
need less evaluations to obtain similarly accurate estimates. Figure 7 illustrates
this, comparing estimation errors |o(c2 ) o(c2 )| (dashed line) and |o(c2 ) o(c2 )|
(full lines) respectively, after x evaluations, for multiple setups with dierent
= 2 1 . Clearly, if is large, the distributions for c1 and c2 do not overlap,
i.e. samples from c1 are not relevant for c2 and o
o. However, the lower , the
greater the overlap. As approaches 0, observations become equally relevant
for both designs and only half the evaluations are needed to obtain similarly
accurate estimates.
Related, yet arguably more important in an optimization setting, is that we
can determine more quickly which of 2 similar designs performs better, i.e. have a
more reliable gradient to guide the search. As similar designs share performance
observations, their estimation errors will be correlated, i.e. the error on rela-
tive performance will be smaller. In the extreme case where distributions fully
Fig. 7. Absolute estimation error Fig. 8. Gradient accuracy

overlap (e.g. = 0), performance estimates are the same. Figure 8 illustrates
this, comparing the fraction of the runs for which o(c1 ) < o(c2 ) (dashed lines)
and o(c1 ) < o(c2 ) (full lines) holds after x evaluations, for dierent . For high
values both perform good, as o(c1 ) o(c2 ). However, for approaching 0,
designs become more similar, and the independent estimate of c1 is frequently
better, even after many evaluations. However, o(c1 ) < o(c2 ) holds, even for small
, after only a few evaluations. In summary, using IS we generally expect to need
less evaluations to solve a given ADP, where at least two or more designs overlap.
Thus far we have discussed why one would use IS estimates, as opposed to
independent sample averages. But how about using regression model predictions
instead? Similar to IS, these allow one to generalize observations beyond the
design used to obtain them, as such improving data-eciency. The main dif-
ference is that IS is model-free. By choosing a regression model, one introduces
model-bias, i.e. makes prior assumptions about what the tness landscape (most
likely) looks like. Clearly, the specic assumptions are model-specic. As para-
metric models (e.g. linear, quadratic) typically impose very strict constraints,
mainly nonparametric models (e.g. Random Forests [5], Gaussian Processes [13])
are used in general ADP settings. While nonparametric models are more exi-
ble, they nonetheless hinge on the assumption that the performance of similar
designs is correlated (i.e. smoothness), which is in essence reasonable, however
the key issue is that this similarity measure is dened in representation space.
Small changes in representation (e.g. a conguration) might result in large per-
formance dierences, while large changes may not.
Using IS estimates, similar designs will also have similar estimates, but rather
than being based on similarity in representation space, they are based on simi-
larity in execution space. In fact, we can derive similarity measures for designs
based on the overlap of their corresponding distributions of executions. This is
interesting in analysis, but can also be used in solving the ADP; e.g. to maintain
the diversity in a population of designs. One way to measure overlap would be
by computing the Bhattacharyya Coecient [4].
To conclude this section, the use of the proposed IS approach does not exclude
the use of regression models. Both approaches are complementary, as dierent
executions might have a similar desirability, something one could learn based on
correlations in performance observations.
5 Theoretical Feasibility
In what follows we will examine whether it is at all possible to compute o,

what information is required for doing so, and whether this information can be
reasonably assumed to be available in a PbO context.
Clearly, we can compute o if we are able to calculate the weights wc (e) for
any e E and c C. From (2) and (3) it follows that the ability to compute
P r(e|c), i.e. the likelihood of generating an execution e, using conguration c, on
x D, is a sucient condition.1 Here, P r can be viewed as encoding how design

decisions aected algorithm execution. In [2] it was argued that this information
is available in a PbO context, and can therefore be exploited without loss of gen-
erality. In the remainder of this section we will briey sketch how P r(e|c) could
be computed in general. Please, bear in mind that the IS approach described
in this paper does not require P r to be computed as described below, and in
specic scenarios it can often be done more time/space/information eciently.
In [2] algorithm design is viewed as a sequential decision problem. Intuitively,
we start execution, leaving design choices open. We continue execution as long
as the next instruction does not depend on the decisions made for any of these.
If it does, a choice point is reached and we must decide which of the possi-
ble instructions (A) to execute next. This process continues until termination.
Our objective is to make these decisions as to maximize the desirability of the
execution f . Any candidate design c C corresponds to a policy of the form
c : A [0, 1], where c (, a) indicates the likelihood of making decision
a in execution context using design c. Here, encodes the features of the
input and execution so far, on which the decision made by any of the designs
of interest (i.e. C) depends. For instance, in our conditional parameter exam-
ple from Sect. 1, choice points correspond to the (rst) use of a parameter, A
to its possible values and each encodes the parameter used. In our SA
example, choice points correspond to the acceptance decisions faced every itera-
tion, A = {accept, reject} and each encodes the magnitude of worsening
proposed and the time elapsed/remaining.
Let (e) be the likelihood of any stochastic events2 that occurred during an
execution e. The likelihood of generating e using c with x D is given by

n
P r(e|c) = D(x) (e) c (i , ai ). (4)
i=1
where x is the input used, ai the decision made in the ith choice point and i
the context in which it was encountered. Computing (4) requires D(x) and (e)
to be known explicitly, which is not always the case in practice. Luckily, it can
be shown (proof
n in [1]) that we can ignore D(x) and (e) in practice, i.e. use
P r(e|c) = i=1 c (i , ai ) instead to compute wc (e), as doing so will make both
nominator, P r(e|c), and denominator, G(e), D(e) (e) times too small, and as
such their ratio, wc (e), correct. In summary, to compute o, the ability to compute
c (c C) and store (a, , f (e)) e E , suces in general.
6 The Proof of Concept

Thus far we have introduced a novel performance estimation technique in
Sect. 3.2, discussed its potential merits in Sect. 4 and established its theoreti-
cal feasibility in Sect. 5. In what follows we wish to convince the reader that it is
1
However, it is not a necessary one. It can be shown (proof in [1]) that the ability to
compute only the relative likelihood PPr(e|c
r(e|c)
) (e E c, c C) suces.
2
In a deterministic setting (e) = 1.
in fact practical. First, in Sect. 6.1, we discuss some challenges encountered when
using IS estimates in practice, in an ADP setting, and how we have addressed
them in the implementation of a Proof of Concept (PoC).
While performance estimation is a key, it is also only one piece of the puzzle.
The ADP is a search problem, i.e. we must search the design space for the design
maximizing performance. In Sect. 6.2 we describe the high-level search strategy
used in our PoC. As a whole, our PoC resembles a simple SMBO-like framework,
using importance sampling estimates, in place of regression model predictions,
to guide its search. To complement this description and facilitate reproduction,
the source code of our PoC is made publicly available.3
6.1 Practical Challenges

Choice of G: As mentioned in Sect. 3.2, we choose a mixture of C as our G
(see (3)). Two issues arise when trying to decide what values (c) should take.
First, C might be extremely large, making (3) expensive to compute. Second,
we want to make (c) dependent on E , e.g. to focus computational eorts on
unexplored/promising parts of the design space. In our PoC, we view G, at any
point, as the distribution that generated E , as opposed to the one that will be
used to generate future executions. To this purpose, we keep track of C the
designs used to generate E and use (c) = n(c)|E | , where n(c) is the number of
executions performed using c. As such, we can at any point in the search process
freely select which design to evaluate next, yet have a consistent estimate for
any c for which lim|E | n(c)
|E | > > 0. This roughly corresponds to what is
known in IS literature as an Adaptive Defensive Mixture approach [7].
Measuring Estimate Quality: In most state-of-the-art congurators, search

is not only guided by performance predictions, but also the accuracy of those pre-
dictions is taken into account. Sample size N and variance 2 are two classical
indicators of the accuracy of a sample average. From the Central Limit Theorem
(CLT) it follows thatgiven a suciently large sample F F, its sample average
2
F N (F), (F N
)
. As such (F N
)
, a.k.a. the Standard Error (SE), can be used
as a measure of uncertainty. As (F) is typically unknown, its sample estimate

(std) is used instead. Sadly, the CLT does not directly apply to IS estimates, and
predicting their expected accuracy has been an active research area. In our PoC,
we use SE as a measure of uncertainty nonetheless, assuming that it is a reasonable
heuristic, despite the lack of theoretical guarantees. However, what is the sample
size, and how to estimate the standard deviation, of an IS estimate? IS estimates
are all based on the same samples E , however not all of these are relevant for all
designs. In fact, a few observations E c E might dominate o(c). In our PoC we
estimate |E c |, a.k.a. the eective sample size, using the formula below
2
( wc (e))
N (c) = eE 2
. (5)
eE wc (e)
3
github.com/Steven-Adriaensen/IS4APE.
As estimate for the standard deviation we use

eE wc (e) (f (e) o(c))
2

std(c) = . (6)
eE wc (e)
which is the standard deviation of the distribution over E , where the relative
likelihood of drawing e E is given by wc (e). Remark that if designs are disjoint,
both (5) and (6) reduce to their sample average equivalents N and std.
Computational Overhead: As discussed before, the use of IS estimates as

described in this paper is benecial in case of overlap and reduces to an ordinary
sample average otherwise. However, computing an IS estimate is computation-
ally more intensive, increasing the overhead. Again, as long as this overhead is
reasonable compared to the cost of an evaluation, performance gains may be
signicant. Naively computing o(c) using (2) would take O(|E ||C |). Further-
more, we need to recompute o(c) every time E and thus G changes. In our
PoC, we memoize and update G(e), which can be done in O(|E |) for any new
e, allowing us to compute o(c) in O(|E |). A hidden constant is the cost of com-
puting P r(e|c), which may be signicant in some ADPs. One solution would be
to memoize them as well. A nal issue is that P r(e|c) might be extremely small,
yet relevant, because P r(e|c ) may be even smaller for all other c . To avoid
underows, one could use log(P r(e|c)) instead.
Algorithm 1. high-level search strategy for our PoC

1: function PoC(x,cinit ,GP,LP,f ,P r,Z,N P rop,P Size,M axEvals)
2: cinc cinit
3: M ISf, P r, ,
4: Cpool
5: for eval = 1 : M axEvals do
6: P equal mixture of GP and LP(cinc )
7: Cprop P with |Cprop | = N P rop
8: cinc arg maxc Cprop {cinc } M .lbZ (c )
9: Cpool arg maxP Size
c Cpool Cprop M .EI(c )

10: c arg maxc Cpool {cinc } M .EI(c )

11: e execute(c,x)
12: M ISf, P r, E {e}, C {c}
13: cinc arg maxc Cpool {cinc } M .lbZ (c )
14: end for
15: return cinc
16: end function
6.2 High-Level Search Strategy

Finally, we describe the high-level search strategy used in our PoC (see
Algorithm 1). When implementing a search strategy we were faced with the
following design choices: Given our observations thus far, which design should
we evaluate next? On which input? What is our incumbent? Decisions made for
all of these are critical in the realization of a framework competitive with the
state-of-the-art. As this paper is about performance estimation, which is largely
orthogonal to these other decisions, doing so was not our main objective; e.g. to
keep it simple our PoC currently only supports optimization on a single input.4
In the remainder of this section we briey discuss the decisions made for the
other two, followed by a detailed description of the high-level search strategy as
a whole.
Choice of Incumbent: Clearly, we would like our incumbent (cinc ) to be the

best design encountered thus far. However, since we only have an estimate of
performance, we also need to take estimate accuracy into account, making it
a multi-objective problem. In our PoC, we measure the uncertainty about the

estimate as u
nc(c) = std(c)
and update the incumbent each time we come
N (c)
across a design c with greater lb Z (c) = o(c) Z u
nc(c), where Z [0, +[ is

a scalarization factor, passed as an argument to the framework (default 1.96).
Z (c) can be seen as an approximate lower bound of the condence interval
Here, lb
for o with z-score Z (default
95% condence interval).
Choice of Design to be Evaluated Next: To decide which design to use

to generate the next execution, we use the Expected Improvement (EI) crite-
rion [11], also used in SMAC. At each point, we attempt to evaluate the design,
which maximizes the Expected (positive) Improvement over our incumbent:

o(c) o(cinc ) o(c) o(cinc )
EI(c) = (o(c) o(cinc )) +u
nc(c) . (7)
unc(c) unc(c)
where and are the standard normal density and cumulative distribution
will be high for designs estimated to perform well and for those
functions. EI
with high estimated uncertainty; and as such this criterion oers an automatic
balance between exploration and exploitation.
Line by Line Description: At line 2 we initialize the incumbent to be the

design cinit passed as starting point to the solver. At line 3 we initialize the IS
model, denoted by M , and dened as a 4-tuple ISf, P r, E , C , where E is
the sample of executions generated thus far and C the bag of designs used to
generate them. Initially E = C = . Remark that M captures all information
required to compute o and u
and lb
nc and as such EI Z.
Each iteration, we consider designs Cpool C as candidates to be evaluated
next. Ideally Cpool = C, however as C might be huge, this may be intractable. On
4
As this input may be a random input generator (e.g. D) we in essence do not lose
any generality here.
the other hand, we often lack the prior knowledge to manually select Cpool C.
Therefore we adapt Cpool dynamically with |Cpool | PSize (default 10). At line 4
we initialize Cpool to be empty. Each iteration, at lines 67, we sample N P rop
(default 100) designs (Cprop ) from P, a distribution over C, to be considered
for inclusion in Cpool . Here, P is an equal mixture of 2 distributions passed as
arguments to the framework, which can be used to inject heuristic information:
Global Prior (GP): A distribution over C, allowing the user to encode prior
knowledge about where good designs are most likely located in C.
Local Prior (LP): A distribution over C, conditioned on cinc , allowing the
user to encode prior knowledge of how cinc can most likely be improved.
Note that our search strategy only interacts with C through these distributions,
making it design representation independent. In particular, it does not assume
them to be congurations, let alone makes any assumptions about the type of
parameters. At line 8 we update cinc to be the design with the greatest lb Z in
Cprop {cinc }. Having generated Cprop , we must decide whether to include them
in Cpool or not. At line 9 we update Cpool as the P Size designs from Cprop Cpool
with maximal EI. At line 10 we select the design to be evaluated next as the
we evaluate this design at line 11 by
design in Cpool {cinc } maximizing EI,
executing it on given input x and we update E , C (i.e. M ) accordingly at
line 12. At line 13, we update cinc w.r.t. the new M . Finally, after M axEvals
iterations, we return cinc at line 15.
7 Experiments
In this section, we briey evaluate the PoC, described previously, experimentally.
We detail our setup in Sect. 7.1 and discuss our results in Sect. 7.2.
7.1 Experimental Setup

Benchmark: We consider micro-benchmark 1 from [2] as ADP (see Fig. 9). It
consists of a simple loop of at most 20 iterations. At each iteration i we encounter
a choice point and have to decide whether to continue the loop (ai = 1) or
not (ai = 0), if we do, we receive a reward ri drawn from N (1, 4), otherwise
execution terminates. The desirability of the execution is simply the sum of these
20
rewards, i.e. f (e) = i=1 ri . The only input is the random generator rng used to
generate the stochastic events during the execution. Every design is represented
as a conguration c (with |c| = 20), where each parameter value ci [0, 1]
indicates the likelihood of continuing the loop at iteration i. Since one of the
optimizers in our baseline (ParamILS) does not support continuous parameters,
we also consider a discretized variant, where the range of each parameter is
{0, 0.1, ..., 0.9, 1} resulting in a total of 1120 congurations.
We used benchmark 1 because it is conceptually simple and the actual per-
20 i
formance can be computed as o(c) = i=1 j=1 cj . Clearly, we have chosen a
benchmark in which the design space is huge, yet is rendered tractable thanks
to the high similarity between designs, making it an ideal use-case for the pro-
posed IS approach. Remark that the optimal design is c : ci = 1, i with
o(c ) = 20. Also, executions are cheap, allowing us to repeat our experiments
multiple times to lter out the noise in our observations. While this bench-
mark exhibits various other specic features, making it trivial to solve, the only
information required/exploited by the IS approach is P r and f , which it treats
as black box functions. In order to compute these, we stored the number of
iterations performed (#it), and the sum of rewards received (sum r), for each
execution. The latter equals f (e), while the former suces to compute P r as
#it
(1 c#it+1 ) i=1 ci 0 #it 19
P r(e|c) = #it
i=1 ci #it = 20
Baseline: We compare the performance of our PoC to that of 2 state-of-the-art

congurators, (Focused) ParamILS [10] and SMAC [9], chosen as representatives
for the use of independent sample averages and regression models respectively.
We perform 1000 independent runs of 10000 evaluations for every framework,
using their default parameter settings and cinit : ci = 0.5, i. For a fair compari-
son we chose the global and local prior (GP, LP) to be uninformative, uniform
and one-exchange distributions respectively. As a consequence, the local and
global search operators used by all 3 frameworks are essentially the same.
7.2 Results and Discussion

Figure 10 shows the (actual) performance of the incumbent obtained by each
framework, after x evaluations, averaged over 1000 runs. Both SMAC and our
PoC were evaluated on both the continuous (full line) and discretized (dashed
line) setup, ParamILS only on the latter.
In the discretized setup we nd that ParamILS nor SMAC found the optimal
solution (c ) within 10000 evaluations, obtaining an average performance of 2
and 6 respectively. Our PoC required only about 10 and 50 evaluations respec-
tively to obtain a similar performance, and the vast majority of the runs found
20
inputs: rng, c; outputs: #it, sum_r; 18
sum_r = 0; choice point 16 o(c*)
i = 1; PoC (disc.)
14
while i 20 do PoC (cont.)
12 SMAC (disc.)
o(inc)
if rng.unif(0,1) < ci then SMAC (cont.)

10 ParamILS
sum_r = sum_r + rng.gaus(1,2);
8
i = i + 1;
6
else
4
break;
2
end if increments f(e) 0
end while 1 10 100 1000 10000
#it = i-1; Evaluations (E')
Fig. 9. Code for benchmark 1 Fig. 10. Results for benchmark 1

c within the 2000 rst evaluations (the worst needed 7573 evaluations). In the
continuous setting both SMAC and our PoC performed (only) slightly worse
than in the discretized setup.
Note that the results for ParamILS are worse than those reported in [2] and
that our PoC performs worse than WB-PURS described therein. This because
only deterministic policies, i.e. ci {0, 1} i, were considered5 in [2], and there-
fore the design space was much smaller (220 vs. 1120 ), yet included c .
Finally, note that the run times diered for all 3 frameworks. A run using our
PoC, ParamILS and SMAC took on average about 5, 20 and 30 min respectively
on our machine. However, as a single evaluation takes virtually no time, compar-
ing optimizers on this benchmark based on actual run times would be unfair, and
bias results towards those optimizers having the lowest overhead (e.g. least IO
operations), which is furthermore very machine dependent. The actual overhead
per evaluation was small for all frameworks (30180 ms), and as such negligible,
as long as an evaluation takes at least a few seconds, which is typically the case
in more realistic ADP settings.
8 Conclusion
In this paper we proposed a novel way of estimating the performance of a family

of related algorithms. Here, we viewed algorithms as distributions of executions;
and algorithm performance as the expectation of some measure of the desirability
of an execution, over this distribution. We showed how Importance Sampling (IS)
can be used to generalize performance observations across algorithms with par-
tially overlapping distributions, amortizing the cost of obtaining them. Finally,
we described the implementation of an actual framework using this technique,
which we validated experimentally.
This framework, while presenting a few interesting ideas of its own (e.g.
the use of prior distributions to inject prior knowledge), is merely a Proof of
Concept (PoC), and misses some features commonly found in state-of-the-art
congurators, such as optimization across a set of training inputs, preliminary
cuto of poor executions and parallel execution support. Extending our PoC to
support these features is a work in progress. We will also look into integrating
IS into existing frameworks, e.g. to automatically exploit the conditionality of
parameters. The latter should be relatively straightforward, as all we need to
do is keep track of which parameters were actually used during an execution
e, and determine whether a conguration c has the correct values for these
(P r(e|c) = 1) or not (P r(e|c) = 0). Finally, we also plan to investigate the use of
IS in an analysis context, where one is interested in approximating the response
surface as a whole, rather than nding its highest peak.
The experimental validation presented in this paper is minimal and is by
no means intended as a proof of the superiority of the proposed approach in
real-world ADP settings. Rather, the crux of this paper was to present a novel
5
As WB-PURS does not support stochastic policies it was not be included as baseline.
and interesting idea in a technically sound manner, and to show that it is at

least possible to implement it in practice. That being said, we see no reason why
we would not be able to improve performance in more realistic ADP settings.
Clearly, the potential gain crucially depends on the design space considered, if
all designs are suciently dierent, we can do no better, but overhead aside, we
see no reason why we would do worse either, as all estimators presented in this
paper, reduce to their sample average equivalents in this case.
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Springer, Dordrecht (1987). doi:10.1007/978-94-015-7744-1 2
Test Problems for Parallel Algorithms
of Constrained Global Optimization
Konstantin Barkalov(B) and Roman Strongin
Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

barkalov@vmk.unn.ru, strongin@unn.ru
Abstract. This work considers the problem of building a class of test

problems for global optimization algorithms. The authors present an
approach to building multidimensional multiextremal problems, which
can clearly demonstrate the nature of the best current approximation,
regardless of the problems dimensionality. As part of this approach, the
objective function and constraints arise in the process of solving an auxil-
iary approximation problem. The proposed generator allows the problem
to be simplied or complicated, which results in changes to its dimen-
sionality and changes in the feasible domain. The generator was tested by
building and solving 100 problems using a parallel global optimization
index algorithm. The algorithms results are presented using dierent
numbers of computing cores, which clearly demonstrate its acceleration
and non-redundancy.
Keywords: Global optimization Multiextremal functions Non-

convex constraints
1 Introduction
One of the general approaches to studying and comparing multiextremal opti-
mization algorithms is based on applying these methods to solve a set of test prob-
lems, selected at random from a certain specially constructed class. In this case,
each test problem can be viewed as a random function created by a special gen-
erator. Using multiextremal optimization algorithms with large samples of such
problems allows the operating characteristics of the methods to be evaluated (the
likelihood of properly identifying the global optimizer within a given number of
iterations), thus characterizing the eciency of each particular algorithm.
The generator for one-dimensional problems was suggested by Hill [1]. These
test functions are typical for many engineering problems; they are particularly
reminiscent of reduced stress functions in problems with multiple concentrated
loads (see [2] for example). Another widely known class of one-dimensional test
problems is produced using a generator developed by Shekel [3].
A special GLOBALIZER software suite [4] was developed to study vari-
ous one-dimensional algorithms with random samples of functions produced by
the Hill and Shekel generators. A comprehensive description of this system, its
https://doi.org/10.1007/978-3-319-69404-7_2
Test Problems for Parallel Algorithms of Constrained Global Optimization 19
capabilities and example uses is provided in [5]. It should be noted that the Hill
functions were successfully used in the design of a one-dimensional constrained
problem generator with controlled measure of a feasible domain [6].
Another generator for random samples of two-dimensional test functions,
successfully used in the studies by a number of authors, was developed and
investigated in [710]. A generator for functions with arbitrary dimensionality
was suggested in [11]. It was used to study certain multidimensional algorithms
as described in [1215]. Well-known collections of test problems for constrained
global optimization algorithms were proposed in [16,17].
All of these generators produce the function to be optimized. In the case
when the dimensionality is greater than two the optimization process itself can-
not be clearly observed. In this regard, it is interesting to examine a dierent
approach, initiated in [5]. In this approach, the objective function appears as
a solution to a certain supporting approximation problem, which allows the
nature of the best current estimate and the nal result to be observed, regard-
less of the number of variables. Complicating the problem statement (including
non-convex constraints) results in an increase of its dimensionality. In fact, the
proposed generator produces an approximation problem to which the objective
function is related.
2 Problem Statement
Lets consider the mathematical model of a charged particle moving through a

magnetic eld along the u axis in the form
mu = eu + F (1)
where m > 0 is the particles mass, u(t) is the particles current position at a
moment of time t 0, eu is an attractive force aecting the particle, F is an
external force applied to the particle along u axis (control action). It is assumed
that control action F is a function of time and is represented as
n

F =m Ai sin(i t + i ).
i=1
Here n > 0 is the dimensionality of the vectors and determines the number of
frequencies in the control action.
Substituting 02 = e/m the Eq. (1) is reduced to a known equation of forced
oscillations
n

u + 02 u = Ai sin(i t + i ). (2)
i=1
The problem is to nd own frequency, control action and initial conditions such
that:
20 K. Barkalov and R. Strongin
1. at t [a, b] the particle would deviate from the position q0 by no more than
> 0;
2. at t = t1 , t2 , t3 , the particle would deviate from positions q1 , q2 , q3 respectively
by no more than > 0;
3. at t = t3 the particle speed would be maximized.
This problem statement can be interpreted as follows: the trajectory of particle
movement u(t) shall pass within a tube, then through the three windows,
with maximum slope in the last of the windows. The illustration in Fig. 1
shows a graph of the function u(t) of the solution to problem (2), which passes
through the tube and windows shown in the chart by dashed lines.
Fig. 1. Problem solution u(t)
Thus, for a particle of a given mass m > 0 it is necessary to determine its

own frequency 0 , the amplitudes Ai , frequencies i and phases i of the control
action, as well as the initial conditions u0 = u(0), u0 = u(0) for the Eq. (2), such
that conditions 13 are true.
Using the following notation for the Eq. (2) solution
n

u(t, , c) = [c2i+1 sin(i t) + c2i+2 cos(i t)] , (3)
i=0
where c = (c1 , . . . , c2n+2 ), = (0 , 1 , . . . , n ), we can represent the original

problem as a constrained maximization problem with parameters c and :
|u(t3 , , c)| max

|u(ti , , c) qi | , i = 1, 2, 3, (4)
|u(t, , c) q0 | , t [a, b].
Solving the optimization problem (4) and nding the vectors , c the solution
to the original problem (2) can be written in accordance with the following
relationships:
n n
u0 = i=0 c2i+2 , u0 = i=0 c2i+1 i
2
Ai = 02 i2 c2i+1 + c22i+2 , 1 i n, (5)
c2i+2
i = arcsin , 1 i n.
c22i+1 +c22i+2
As follows from formula (3), the parameters c are included in the equation
solution linearly, and the parameters non-linearly. Given the constraints (4)
this allows the problem to be reformulated and c to be found without using a
numerical optimization method.
Lets consider a set of points (j , uj ), 0 j m, with the coordinates dened
as follows:
j = a + jh, uj = q0 , 0 j m 3, (6)
m2 = t1 , um2 = q1 ,
m1 = t2 , um1 = q2 ,
m = t 3 , u m = q3 ,
where h = (b a)/(m 3), i.e. the rst m 3 points are located at equal
distances in the center of the tube, the other three align with the centers of
the windows.
The requirement is that the trajectory of particle u(t) passes near the
points (j , uj ), 0 j m. If the measure of deviation from the points is dened
as the sum of the squared deviations
m
2
(c, ) = [uj u(j , , c)] ,
j=0
then the parameters c can be found (given xed values of ), by solving the least
squares problem
c () = arg min (c, ). (7)
According to the least squares method, the solution to problem (7) can be
obtained by solving a system of linear algebraic equations regarding the unknown
c, which can be done, e.g., by Gaussian elimination.
It should also be considered that the components of frequency vector can
be placed in ascending order, so as to avoid duplicate solutions corresponding to
the vector with similar components in dierent order. In addition, it is natural
to assume that frequencies in the control action must not just be ordered but
dier by a certain positive value, as an actual physical device can only generate
control signals with a certain precision. This assumption can be represented in

the form of a requirement for the following inequalities to be true:
i1 (1 + ) i (1 ) 0, 1 i n. (8)
Here (0, 1) is a parameter reecting the precision of signal generation by the

control device.
Then the original problem can be reformulated as follows:
= arg max |u(t3 , , c ())| (9)

i1 (1 + ) i (1 ) 0, 1 i n,
|u(ti , , c ()) qi | , i = 1, 2, 3,
max u(t, , c ()) min u(t, , c ()) ,
t[a,b] t[a,b]
where c () is determined from (7), and the number of constraints will be depen-
dent on the number of frequencies n in the control action.
Figure 1 shows trajectory u(t), which corresponds to the solution to problem
(9) with parameters a = 1, b = 10, t1 = 13, t2 = 16.65, t3 = 18, q0 = q3 = 0,
q1 = 7.65, q2 = 9.86. The solution (with three signicant digits)
u(t) = 34.997 sin(0.01t 0.061) + 11.323 sin(0.902t 0.777)

19.489 sin(1.023t + 0.054) + 9.147 sin(1.139t + 0.633)
is determined by the optimal vectors = (0.01, 0.902, 1.023, 1.139) and c =

(34.93, 2.151, 8.075, 7.936, 19.461, 1.048, 7.375, 5.411). As follows from (5),
the original problem with the solution u(t) is noted as
u + 104 u = 9.213 sin(0.902t 0.777) 20.383 sin(1.023t + 0.054)

11.842 sin(1.139t + 0.633),
u0 = 3.62, u0 = 4.556.
3 Generating a Series of Problems
The numeric experiments described below used a generator based on the approx-
imation problem from Sect. 2. Apparently, the variation in any parameter of the
original problem (2) will change the optimization problem (9), so it is sucient
to vary just a few of them.
The centers of the rst two windows, i.e. the pairs (t1 , q1 ) and (t2 , q2 ),
were chosen as the parameters for determining the specic problem statement.
The values q1 and q2 were chosen independently and uniformly from the ranges
[1, 10] and [10, 1], respectively. The values t1 and t2 were dependent: rst, the
value t1 was chosen from the range [b + 1, t3 2], then, the value t2 was chosen
from the range [t1 + 1, t3 1]. All other parameters in problem (2) were xed:
a = 1, b = 10, t3 = 18, q0 = q3 = 0, = 0.3. Parameter from (8) was chosen
at 0.05. The number of points in the additional grid (6) for solving the least
square problem (7) was set at 20. The problem of one-dimensional maximization
and minimization from (9) were solved by a scanning over a uniform grid of 100
nodes within the interval [a, b].
An important feature determining the existence of a feasible solution for the
problem being considered is the number and range of frequency variation in the
vector . If the range is too small, or the number of frequencies is insucient,
the feasible domain in the problem (9) will be empty. In the experiments carried
out, the number of frequencies was chosen to be n = 3, which corresponds to
= (0 , 1 , 2 , 3 ), while the variable frequency change range was set from 0.01
to 2, i.e. i [0.01, 2], 0 i 3.
Lets note some important properties of the problems produced by the gen-
erator under consideration.
Remark 1. Problem constraints (9) are dierent in terms of the time required
to verify them. For example, checking each of the rst n constraints in (9) (lets
call these constraints geometric)
i1 (1 + ) i (1 ) 0, 1 i n,
requires performing only three operations with real numbers. Testing other con-
straints (we will call them the main constraints)
|u(ti , , c ()) qi | , i = 1, 2, 3,
max u(t, , c ()) min u(t, , c ())

t[a,b] t[a,b]
is far more labor-intensive. First, this requires producing a system of linear

equations to solve the problem (7) (m(2n + 2)2 computing of sin and cos).
Second, this system needs to be solved ( 23 (2n+2)3 operations on real numbers).
Third, two one-dimensional optimization problems need to be solved at the last
constraint in (9) (100 computing of sin and cos).
Remark 2. The problems are characterized by multiextremal constraints,

which form a non-convex feasible domain. For example, Fig. 2 show level lines
for the functions in the right-hand sides of the main constraints for one of the
problems, while Fig. 3 shows level lines of the objective function. These lines are
provided within the feasible domain of the geometric constraints.
Fig. 2. Level lines for the main constraints
Fig. 3. Level lines for the objective function

Remark 3. Increasing the frequency change range results in new solutions

appearing in an area of higher frequencies; the feasible domain of the optimiza-
tion problem becomes multiply connected. For example, Fig. 4 shows two trajec-
tories, the solid line corresponds to the vector = (0.01, 0.902, 1.023, 1.139) and
solution
u(t) = 34.997 sin(0.01t 0.061) + 11.323 sin(0.902t 0.777)

19.489 sin(1.023t + 0.054) + 9.147 sin(1.139t + 0.633)
obtained at i [0.01, 2], 0 i 3, dashed line corresponds to the frequency

vector = (1.749, 1.946, 2.151, 2.377) and solution
u(t) = 5.434 sin(1.749t + 0.832) + 12.958 sin(1.946t + 0.241)

+ 11.844 sin(2.151t 1.377) + 4.302 sin(2.377t + 0.501)
obtained at i [0.01, 4], 0 i 3. Obviously, the second solution is better, as

the trajectory at the last point has the largest slope.
Fig. 4. Solutions with dierent frequencies
These optimization problem properties allow us to conclude that this gener-

ator may be applied for testing parallel global optimization algorithms. Inter-
ested readers may nd the review of approaches to parallelization of optimiza-
tion algorithms in [18]. In this study we will use a parallel algorithm based on
information-statistical approach [5,19].
4 Parallel Global Optimization Index Algorithm

Lets consider a multiextremal optimization problem in the form
(y ) = min {(y) : y D, gj (y) 0, 1 j m}, (10)
where the search domain D is represented by a hyperparallelepiped

D = y RN : ai yi bi , 1 i N . (11)
Suppose, that the objective function (y) (henceforth denoted by gm+1 (y)) and
the left-hand sides gj (y), 1 j m, of the constraints satisfy Lipschitz condi-
tion
|gj (x1 ) gj (x2 )| Lj y1 y2 , 1 j m + 1, y1 , y2 D.
with respective constants Lj , 1 j m + 1, and may be multiextremal. Then,

it is suggested that even a single computing of a problem function value may
be a time-consuming operation since it is related to the necessity of numerical
modeling in the applied problems (see, for example, [20,21]).
Using a continuous single-valued mapping y(x) (Peano-type space-lling
curve) of the interval [0, 1] onto D from (11), a multidimensional problem (10)
can be reduced to a one-dimensional problem
(y(x )) = min {(y(x)) : x [0, 1], gj (y(x)) 0, 1 j m}, (12)
The reduction of dimensionality matches the multidimensional problem with

a Lipschitzian objective function and Lipschitzian constraints with a one-
dimensional problem where the respective functions satisfy the uniform Holder
condition (see [5]), i.e.
1/N
|gj (y(x1 )) gj (y(x2 ))| Kj |x1 x2 | , x1 , x2 [0, 1], 1 j m + 1.
Here N is the dimensionality of the original multidimensional problem, and the

coecients Kj are related to Lipschitz constants Lj with formulae

Kj 2Lj N + 3, 1 j m + 1.
The issues around the numeric construction of a Peano-type curve and the cor-
responding theory are considered in detail in [5,19]. Here we can just state that
the numerically computed curve (evolvent) is an approximation of the theoreti-
cal Peano curve with a precision at least 2m for each coordinate (the parameter
m is called the evolvent density).
Lets introduce the classication of points x from the search domain [0, 1]
using the index = (x). This index is determined by the following conditions:
gj (y(x)) 0, 1 j 1, g (y(x)) > 0,

where the last inequality is negligible if = m + 1, and meets the inequalities

1 = (x) m + 1. This classication produces a function
f (y(x)) = g (y(x)), = (x),
which is determined and computed along the interval [0, 1]. Its value in a point
x is either the value of the left part of the constraint violated at this point (in
the case, when m), or the value of the objective function (in the case, when
= m + 1). Therefore, determining the value f (y(x)) is reduced to a sequential
computation of the values
gj (y(x)), 1 j = (x),
i.e. the subsequent value gj+1 (y(x)) is only computed if gj (y(x)) 0. The com-
putation process is completed either when the inequality gj (y(x)) > 0 becomes
true, or when the value of (x) = m + 1 is reached.
The procedure called trial at point x automatically results in determining
the index for this point. The pair of values
z = g (y(x)), = (x), (13)
produced by the trial in point x [0, 1], is called the trial result.
A serial index algorithm for solving one-dimensional conditional optimization
problems (12) is described in detail in [6]. This algorithm belongs to a class of
characteristical algorithms (see [7]). It can be parallelized using the approach
described in [7] for solving unconstrained global optimization problems. Lets
briey describe the rules of the resulting parallel index algorithm (PIA).
Suppose we have p 1 computational elements (e.g., CPU cores), which
can be used to run p trials simultaneously. In the rst iteration of the method,
p trials are run in parallel at various random points xi (0, 1), 1 i p.
Suppose n 1 iterations of the method have been completed, and as a result
of which, trials were carried out in k = k(n) points xi , 1 i k. Then the
points xk+1 , . . . , xk+p of the search trials in the next (n + 1)-th iteration will be
determined according to the rules below.
1. Renumber the points x1 , . . . , xk from previous iterations with lower indices,

lowest to highest coordinate values, i.e.
0 = x0 < x1 < . . . < xi < . . . < xk < xk+1 = 1, (14)
and match them with the values zi = g (y(xi )), = (xi ), 1 i k,

from (13), calculated at these points; points x0 = 0 xk+1 = 1 are introduced
additionally; the values z0 zk+1 are indeterminate.
2. Classify the numbers i, 1 i k, of the trial points from (14) by the number
of problem constraints fullled at these points, by building the sets
I = {i : 1 i k, = (xi )} , 1 m + 1, (15)
containing the numbers of all points xi , 1 i k, with the same values of

. The end points x0 = 0 and xk+1 = 1 are interpreted as those with zero
indices, and they are matched to an additional set I0 = 0, k + 1.
Identify the maximum current value of the index
M = max { = (xi ), 1 i k} . (16)
3. For all values of , 1 m + 1, calculate the values

|zi zj |
= max 1/N
: i, j I , j < i . (17)
(xi xj )
If the set I contains less than two elements or from (17) equals zero, then
assume = 1.
4. For all non-empty sets I , 1 m + 1, determine the values

, < M,
z = (18)
min {g (xi ) : i I }, = M,
where M is the maximum current value of the index, and the vector
R = ( 1 , . . . , m ) , (19)
with positive coordinates is called the reserve vector and is used as a para-
meter in the algorithm.
5. For each interval (xi1 , xi ),1 i k + 1, calculate the characteristic R(i):
(zi zi1 )2 zi + zi1 2z
R(i) = i + 2 , = (xi1 ) = (xi ),
(r )2 i r
zi z
R(i) = 2i 4 , (xi1 ) < (xi ) = ,
r
zi1 z
R(i) = 2i 4 , = (xi1 ) > (xi ).
r
where i = (xi xi1 )1/N , and the values r > 1, 1 m + 1, are used
as parameters in the algorithm.
6. Reorder the characteristics R(i), 1 i k + 1, from highest to lowest
R(t1 ) R(t2 ) . . . R(tk ) R(tk+1 ) (20)
and choose p largest characteristics with interval numbers tj , 1 j p.
7. Carry out p new trials in parallel at the points xk+j , 1 j p, calculated by
the formulae
xtj +xtj 1
xk+j = 2 , (xtj 1 ) = (xtj ),
N
xtj +xtj 1 sign(ztj ztj 1 ) |ztj ztj 1 |
xk+j = 2 2r , (xtj 1 ) = (xtj ) = .
The algorithm stops if the condition tj becomes true for at least one
number tj , 1 j p; here > 0 has an order of magnitude of the desired
coordinate accuracy.
Lets formulate the conditions for algorithm convergence. For this, in addition
to the exact solution y of the problem (10), we will also consider the -reserved
solution, determined by the conditions
(y ) = min {(y) : y D, gj (y) j , 1 j m},
where 1 , . . . , m are positive numbers (reserves for each constraint). Lets also
introduce the set
Y = {y D : gj (y) 0, (y) (y )} (21)
of all feasible points for the problem (10), which are no worse (in terms of the
objective functions value) than -reserved solution.
Using this notation, the convergence conditions can be formulated as the
theorem below.
Theorem. Suppose the following conditions are true:
1. The problem (10) has -reserved solution y .

2. Each function gj (y), 1 j m + 1, satises Lipschitz condition with the
respective constant Lj .
3. The parameters j , 1 j m, from (19) have the values of the respective
coordinates from the reserve vector R .
4. For the values from (17) starting from a certain iteration, the following
inequalities are true:

r > 231/N L N + 3, 1 m + 1.
5. Each iteration uses a xed number of computational elements p, 1 < p < ,

and each trial is completed by a single computational element within a nite
time.

Then any accumulation point y of the sequence y k generated by parallel
index method while solving the problem (10) is admissible and satises the
conditions

(y) = inf (y k ) : gj (y k ) 0, 1 j m, k = 1, 2, . . . (y ).
i.e., y belongs to the set Y from (21).

These convergence conditions proceed from the theorem of convergence of
the serial index algorithm [5] and the theorem of convergence of a synchronous
characteristical algorithm [7].
5 Results of Numerical Experiments

The procedure applied to evaluate the eciency of the algorithm uses an oper-
ating characteristics method, originally described in [22], which consists of the
following.
Suppose a problem from the series under consideration be solved by acertain
algorithm. The problem is associated with the sequence of trial points y(xk )
produced by the algorithm. The sequence is truncated either when a trial point
falls (for the rst time) into a certain -vicinity of the solution y or when a
certain number of trials K does not produce such a point. In our experiments
we used K = 106 .
The results obtained by solving all problems in a series by means of the
algorithm is represented by the function P (k) dened as the fraction of problems
in which some trial point fall within the given -neighborhood of the solution
in the rst k steps. This function is called the operating characteristic of the
algorithm.
Since the specication of an -vicinity requires that y be known, its value
was estimated in advance (for each problem) by searching through all nodes of a
uniform grid (dened on the search domain), e.g. with a step 102 by coordinates.
As discussed above (see Remark 1 from Sect. 3), the constraints in the prob-
lem being addressed have a dierent nature. The rst three constraints are geo-
metric, and checking whether they are feasible is not hard. Checking other con-
straints is far more labor-intensive. Therefore, when building operating charac-
teristics for this class of problems, we will only consider the trials that resulted
in checking labor-intensive constraints. The trials that were completed while
checking geometric constraints are not included in the total number of trials.
The experiments were carried out for the parallel index algorithm (PIA)
described above, with the number of used cores p varying from 1 to 8. The
parameters used were r = 2.5, 1 8. Components of the reserve vector
from (19) were selected adaptively under the rule = 0.005 , 1 7,
where is from (17) (see [5] for a justication of this component selection
for the reserve vector). Computing experiments were carried out on one of the
nodes of a high-performance cluster of the Nizhny Novgorod State University.
The cluster node includes Intel Sandy Bridge E5-2660 2.2 GHz CPU and 64 Gb
RAM. For implementation of the parallel algorithm OpenMP was used.
The algorithms operating characteristics when using dierent number of
computing cores p obtained with a series of 100 problems, are shown in Fig. 5.
The location of the curves shows that any of the parallel algorithms solve 100%
of the problems in less than 7 105 trials, with more than 80% of the problems
completing within 2 105 trials. The operating characteristics also show that the
algorithm is non-redundant the number of trials in the parallel algorithm does
not grow (compared to serial algorithm) when additional cores are employed.
Now lets evaluate the speedup achieved by using a parallel index algorithm,
depending on the number p of computing cores used. Table 1 shows the aver-
age number of iterations n(p) performed by the algorithm when solving a series
of 100 problems, and speedup by the iterations s(p) of a parallel algorithm.
Fig. 5. Operating characteristics of PIA, which uses p cores
Table 1. Speedup of the algorithm
p n(p) s(p)
1 241239
2 94064 2.56
4 45805 5.27
8 22628 10.66
The results show that the speedup is greater than the number of cores used
(hyper-acceleration). This situation is explained by the fact that the algorithm
performs an adaptive evaluation of the behavior of the objective function (calcu-
lating the lower bounds for the Lipschitz constant (17) and the current minimum
value (18)). For example, if the Lipschitz constant is better estimated in a paral-
lel version, then the parallel algorithm using p cores can be accelerated by more
than p times.
6 Conclusion
In summary, we must note that the method proposed in this work to generate
multidimensional conditional global optimization problems allows:
clear visualization of the best current estimate and the nal solution to the
problem, regardless of the number of variables;
increased dimensionality of the optimization problem being addressed by
varying the original approximation problem;
control of the feasible domain by adding extra non-convex constraints.
The functions included in the optimization problem are computationally inten-

sive, which also dierentiates the mechanism proposed from other known mech-
anisms.
The proposed generator was used to build and subsequently solve 100 prob-
lems using a parallel index algorithm. The operating characteristics of the par-
allel algorithm have been built, clearly demonstrating its non-redundancy.
Acknowledgments. This study was supported by the Russian Science Foundation,

project No. 16-11-10150.
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Automatic Conguration of Kernel-Based
Clustering: An Optimization Approach
Antonio Candelieri1(&), Ilaria Giordani1, and Francesco Archetti1,2

1
Dipartimento di Informatica Sistemistica e Comunicazione, DISCo,
Universit di Milano Bicocca, 20126 Milan, Italy
antonio.candelieri@unimib.it
2
Consorzio Milano Ricerche, via R. Cozzi 53, 20125 Milan, Italy
Abstract. This paper generalizes a method originally developed by the authors

to perform data driven localization of leakages in urban Water Distribution
Networks. The method is based on clustering to perform exploratory analysis
and a pool of Support Vector Machines to process on line sensors readings. The
performance depends on certain hyperparameters which have been considered as
decision variables in a sequential model based optimization process. The
objective function is related to clustering performance, computed through an
external validity index dened according to the leakage localization goal. Thus,
as usual in hyperparameters tuning of machine learning algorithms, the objective
function is black box. In this paper it is shown how a Bayesian framework offers
not only a good performance but also the flexibility to consider in the opti-
mization loop also the automatic conguration of the algorithm. Both Gaussian
Processes and Random Forests have been considered to t the surrogate model
of the objective function, while results from a simple grid search have been
considered as baseline.
Keywords: Hyperparameters optimization Sequential model based

optimization Kernel based clustering Leakage localization
1 Introduction
In this paper we consider a machine learning model based on clustering and Support
Vector Machine (SVM) classication, and its application to leakage localization in an
urban water distribution network.
Machine learning based approaches for analytically localizing leaks in a Water
Distribution Network (WDN) have been proposed, mostly based on the idea that leaks
can be detected by correlating actual modications in flow and pressure within the
WDN to the output of a simulation model whose parameters are set to evaluate the
effect induced by a leak in a specic location and with a specic severity [16].
This paper is based on an analytical framework that uses: (1) extensive simulation
of leaks for data generation; (2) kernel-based clustering to group leaks implying similar
variations in pressure and flow; (3) classication learning (i.e. SVM) to discover the
relation linking variations in pressure and flow to a limited set of probably leaky pipes.

https://doi.org/10.1007/978-3-319-69404-7_3
Automatic Conguration of Kernel-Based Clustering 35
The main contribution of this paper is the formulation of the algorithm congu-
ration problem and a proposal of a global optimization strategy to solve it.
Automatic conguration of machine learning algorithms has been attracting a
growing attention [7]: the use of default parameters, as pointed out in [8], can result in a
poor generalization performance. Grid search, the most widely used strategy for
hyperparameters optimization, is hardly feasible for more than 2 or 3 parameters. For
these reasons optimization methods have been widely proposed. Classical optimization
cannot be used as the performance measures are typically black-box functions, and/or
multimodal, whose derivatives are not available. For these reasons, global optimization
is now widely accepted as the main computational framework for hyperparameters
optimization and algorithm conguration.
Global optimization [911] methods fall in 3 large families. The rst is Partitional
Methods which offer global convergence properties and guaranteed accuracy estima-
tion of the global solutions, e.g., in the case of Lipschitz global optimization [12, 13].
A signicant application of these methods to hyperparameter optimization is given, e.g.
in [14] for the case of SVM regression or in [15] for the case of signal processing. The
second family is Random Search [1618] and the related metaheuristics like simulated
annealing, evolutionary/genetic algorithms, multistart&clustering, largely applied in
global optimization. Random Search has recently received fresh attention from the
machine learning community and increasingly considered as a baseline for global
optimization as in Hyperband [19, 20] which considers randomly sampled congura-
tions and relies on a principled early stopping rule to evaluate each conguration and
compares its performance to Bayesian Optimization (BO).
BO represents the third family which came to dominate the landscape of hyper-
parameter optimization in the machine learning community [2123]. First proposed in
[24], BO is a sequential model based approach: its key ingredients are a probabilistic
model which captures our beliefs given the evaluations already performed and an
acquisition function which computes the utility of each candidate point for the next
evaluation of f. The Bayesian framework is particularly useful when evaluations of f are
costly, no derivatives are available and/or f is nonconvex/multimodal.
Very relevant to the authors activity is that the BO can be applied to unusual design
spaces which involve categorical or conditional variables. This capability makes BO the
natural solution when not only the hyperparameters but also the specic algorithm itself
to be automatically selected in the so called algorithmic conguration [25].
The rest of the paper is organized as follows: Sect. 2 describes the hydraulic
simulation model, the general algorithmic structure and the formulation of the per-
formance measure of the clustering. Section 3 is devoted to the sequential model based
optimization for the hyperparameters optimization and the description of the software
environment utilized. Section 4 analyzes the computational performance of different
strategies while Sect. 5 contains concluding remarks.
36 A. Candelieri et al.
2 Material and Methods
In this paper the reference application domain is an urban WDN. Water network design
and optimization of the operations (pump scheduling) has received a lot of attention in
the Operation Research literature [26].
The elements of the network are subject to failures, not uncommon given the
typically old age of the infrastructure. Breakages of pipes, in particular, generate bursts
and leaks which can inhibit supply service of the network (or a subnetwork) and induce
substantial loss of water, with an economic loss (no-revenue water), water quality
problems and unnecessary increase in energy costs.
The state of the WDN is usually monitored by a number of sensors which record
flows and pressures. When a leak occurs, sensors record a variation from normal
operating values. We named the vector of deviations the signature of the leak.
The main aim of the proposed machine learning approach is to use this signature
to identify the location of the leak. Therefore, the basic idea is to move between the
physical space (pipes of the WDN) and the space of leak signatures to infer a
possible relation both direct and inverse between causes (leaks on pipes) and effects
(variations in flow and pressure at the monitoring points).
Although data could be gathered looking at historical leakage events, they would be
too sparse and of poor quality. Thus, a hydraulic simulation software is used to gen-
erate a wide set of data emulating the data from sensors according to different leakage
scenarios in the WDN, consisting in placing, in turn, a leak on each pipe and varying
its severity in a given range. Our choice of simulator is EPANET 2.0, widely used for
modeling WDNs and downloadable for free from the Environmental Protection
Agency web site (http://www.epa.gov/nrmrl/wswrd/dw/epanet.html).
In this paper we focus on optimizing, throughout Sequential Model Based Opti-
mization (SMBO) [27], a set of design variables which are hyperparameters of a
machine learning based system which, given a new leak signature, infers its location as
a limited set of probably leaky pipes. Learning is performed on a dataset obtained as
vectors of N components, where N is the overall number of sensors (N = Np + Nf, with
Np = number of pressure sensors and Nf number of flow sensors), that is the Input
Space. According to the Fig. 1, learning is performed in two stages: one unsupervised,
aimed at grouping together similar signatures to reduce the number of different
effects, and one supervised, aimed at estimating the group of signatures which the
signature of a real leak belongs to. This allows for retrieving only the scenarios related
to the signatures in that cluster and, therefore, leaky pipes associated to those
signatures.
The basic idea of the analytical framework has been presented in [2830], where
Spectral Clustering (SC) is used for the unsupervised learning phase and Support
Vector Machine (SVM) classication is used for the supervised learning phase. The
cluster assignment provided to each instance (i.e. signature) is used as label to train the
SVM classier. While the clustering is used to model the direct relation from leak
scenario (i.e. leaky pipe and leak severity) to a group of similar signatures, the SVM
inverts this relation. Thus, when a reading from sensors is acquired, the variations with
respect to the faultless WDN model are computed and the resulting signature is given
Fig. 1. Overall machine learning based leakage localization system
as input to the trained SVM which assigns the most probable cluster the signature
belongs to. Finally, the pipes relative to the scenarios in that cluster are selected as
leaky pipes.
Although SC proved to be effective, it is computationally expensive and, in this
study, we propose to replace SC with a Kernel k-means algorithm and apply SMBO to
optimally tune its hyperparameters. This allows to reduce the computational burden
induced and to infer the similarity measure even non-linear directly from data and
implicitly through the kernel trick, instead to dene a priori a similarity measures for
weighting edges of the afnity graph. It has been already reported in literature that an
equivalence exists between SC and kernel k-means [31]. In particular, a Radial Basis
Function (RBF) kernel has been chosen, characterized by its hyperparameter r. The
other hyperparameter is, naturally, the optimal number k of clusters.
Moreover, it is important to highlight that adoption of RBF kernels in global
optimization algorithms has been also investigated, and compared with statistical
models [32].
2.1 Notation
The WDN can be represented through a graph: G hVjE i, where:
V is the set of junctions, which are consumption points, reservoirs, tanks, emitters
as well as simple junctions between two pipes;
E is the set of links, which are pipes, pumps or valves.
Let us denote with

E
E
the set of pipes within the set of links. Furthermore, let us denote with
A fa1 ; . . .; al g
the set of severities of the (simulated) leaks. Then, the set of Leakage Scenarios
can be dened as follows:
A
SE
with severity a 2 A.
where se;a 2 S is related to a leak on the pipe e 2 E
Finally, the signature of a leak, that is the effect induced by specic leakage
scenario, is dened as:

we;a f se;a
with f(.) a function to compute, through EPANET, the expected variations of pressure
and flow at the sensors locations.
Clustering is performed on signatures, therefore every cluster Ck is a set of
signatures (i.e. similar effects due to different leaks):

a2A :
Ck xe;a 2 RN ; xe;a f se;a ; e 2 E;
It is important to highlight that the clustering procedure used is crispy, so
C i \ C j ; 8i 6 j:
Many clustering quality measures are given in the literature [33], divided between
internal basically related to inter- and intra- distances and external which
need a set of labeled examples and treat the clustering as a classication problem.
External measures are domain specic therefore more effective in our case but they
cannot be used because as remarked above historically are sparse, unbalanced and poor
quality. For these reasons we have dened ad hoc composite index to address the
leakage localization objective. In particular, to obtain an effective and efcient local-
ization of possible leaks into a WDN, clusters have to satisfy the following properties:
The set of pipes candidate as leaky must be as limited as possible, for every cluster;
The signatures of scenarios associated to a given pipe should be spread as less as
possible among different clusters.
We refer to the rst property as compactness and to the second one as a proxy of
accuracy.
Before to present the nal index, the two following sets have to be dened:

k e 2 E
E : 9a 2 A : xe;a f se;a 2 C k
and

: xe;a f se;a 2 C k :
Sk;a se;a 2 S; e 2 E
The index for evaluating the tness of clustering is the composition of two different
measures:
IC IP
I
2
where IC measures the compactness of the cluster in terms of number of pipes
identied as probably leaky with respect to the overall number of pipes in the WDN,
while IP measures a sort of accuracy that the leak is in the set of pipes identied as
leaky instead to be on other pipes.
More in detail the two measures are computed as follows:
PK
k k

k1 IC E
IC PK
k
k1 jE j
where
k
j E
jE
ICk
jE j 1
and
IP avgk IPk
where
P
1 k;a
j Aj a2A S
IPk kj :
jE
Let us suppose that Ck contains only signatures associated to all the scenarios
then:
related to only one pipe e 2 E,
k
E 1, because E
k fe g;
k;a
S 1, because 8a 2 A; Sk;a s 2 S : 9!e 2 E : x f se ;a 2 C k ;
1
j Aj
IPk j AjEj k j 1.
On the other side, let us suppose that Ck contains signatures associated to |A|
different pipes with |A| different severities, then:
k
E j Aj, for the hypothesis;
k;a
S 1, because 8a 2 A; Sk;a s 2 S : 9!e 2 E : x f se ;a 2 C k ;
P k;a
a2A S j Aj;
1
j Aj
IPk j AjEj k j j A1 j \1.
k k
In general, E can be greater than j Aj; in the worst case E
j Aj, thus IPk 1.
2.2 The Case Study and the Data Generation Process

In the following Fig. 2 the specic WDN considered in this study is reported. It is a
District Metered Area (DMA), namely Neptun, of the urban WDN in Timisoara,
Romania, and was a pilot of the European Project ICeWater.
Fig. 2. The District Metered Area (DMA) Neptun of the urban WDN in Timisoara, Romania:
the WDN case study considered in this paper
Neptune consists of 335 junctions (92 are consumption points) and 339 pipes. In
the proposed approach, EPANET is used to simulate a wide set of leakage scenarios,
consisting in placing, in turn, a leak on each pipe and varying its severity in a given
range.
At the end of each run, EPANET provides pressures and flows at each junction and
pipe, respectively, and, therefore, also at the monitoring points (i.e. sensors). Variations
in pressure and flow induced by a leak (i.e. signature of the leak) are computed with
respect to the simulation of the faultless WDN. Finally, a dataset is obtained having as
many instances as number-of-pipelines times number-of-discharge-coefcient-values.
2.3 Kernel K-means

The kernel k-means algorithm [34] is a generalization of the standard k-means algorithm.
By implicitly mapping points to a higher-dimensional space, kernel k-means can
discover clusters that are non-linearly separable in input space. This provides a major
advantage over standard k-means, and allows us to cluster points if we are given a
positive denite matrix of similarity values.
A general weighted kernel k-means objective is mathematically equivalent to a
weighted graph partitioning objective [31]; this equivalence has an important conse-
quence: in cases where eigenvector computation is prohibitive, kernel k-means elim-
inates the need for any eigenvector computation required by graph partitioning.
Given a set of vectors x1, x2, , xn, the kernel k-means objective can be written as a
minimization of:
K X
X
kUxi xk k2
k1 x2C k
where U x is a (non-linear) function mapping vectors xi from the Input Space to the
Feature Space, and where xk is the centroid of cluster Ck.
Expanding kUxi xk k2 in the objective function, one can obtain:
P P
2 xj 2C k Uxi U xj xj ;xi 2C k U xj Uxi
Uxi Uxi :
jC k j jCk j2
Therefore, only inner products are used in the computation of the Euclidean dis-
k
tance between every vector and the centroid
of the cluster C . As a conclusion, given a
kernel matrix K, where Kij = Uxi U xj , these distances can be computed without
knowing explicit representations of U x.
3 Hyperparameter Optimization of the Unsupervised

Learning Phase of the Machine Learning Pipeline
Although the overall analytical leakage localization is composed of two learning stages,
the focus of this paper is on the optimization of hyperparameters of the rst unsu-
pervised learning phase (i.e. Kernel k-means clustering).
Clustering of leak signatures is aimed at grouping together similar effects induced
by different (simulated) leaks. This allows to implement the second supervised learning
phase considering a limited number of labels (i.e. the number of clusters instead of the
number of pipes of the WDN). Previous Fig. 1 summarizes the overall machine
learning based leakage localization approach where Kernel k-means clustering
replaces Spectral Clustering used in the preliminary papers.
3.1 Hyperparameters in the Pipeline: The Design Variables

The number and type of hyperparameters in the overall machine learning pipeline
depends on the specic software used for kernel-based clustering and SVM classication.
Since the focus of this paper is to replace the original SC phase with a kernel-based
k-means, just the two following hyperparameters are taken into account:
The number k of clusters a discrete decision variable (i.e. an integer);
The value of the RBF kernels parameter r a continuous decision variable.
The SVM hyperparameters are not part of the optimization process in this paper.
3.2 Clustering Performance: The Objective Function

The objective function, that is the clustering performance index I dened in previous
Sect. 2.1, is black-box. To compute it, the execution of the kernel k-means is per-
formed on the entire dataset, given k and r, along with the calculation of IC and IP.
3.3 Sequential Model Based Optimization

The generic Sequential Model Based Optimization (SMBO) [27] consists in:
1. starting with an initial set of evaluations of the objective function;
2. tting a regression model (i.e. surrogate of objective function) based on the overall
set of evaluations performed so far;
3. querying the regression model to propose a new, promising point, usually through
the optimization of an acquisition function (or inll criterion);
4. evaluating objective function at the new point and adding the results to the set of
evaluations;
5. if a given termination criterion (e.g. maximum number of evaluations of the
objective function) is not satised, then return to step 2.
Several adaptations and extensions, e.g., multi-objective optimization [36], multi-
point proposal [37, 38], more flexible regression models [27] or alternative ways to
calculate the inll criterion [39] have been investigated recently, as reported in [40].
A specic application domain for SMBO is the hyperparameter optimization for
machine learning algorithms [4143], where the algorithm is a black-box and the
objective function is one or multiple performance measure(s).
3.3.1 Building the Surrogate of the Objective Function: Gaussian

Processes and Random Forest
Generation of a surrogate of the objective function is one the rst choices in designing
a SMBO process. Selection of the specic regression model to use can be suggested by
the search space spanned by the decision variables. Kriging [35], based on Gaussian
Processes (GP), is usually recommended, but, in the case the search space also includes
categorical parameters, Random Forests (RF) are a plausible alternative [27] as they
can handle such parameters directly.
In this study, two hyperparameters are considered: the number k of clusters which
is an integer categorical variable and r, the internal parameter of the kernel clustering
which is a continuous variable. Although RF should be preferable, due to the nature of
the hyperparameter k, also GP has been investigated in this study. The initial design
consists of 220 evaluations obtained through Latin Hypercube Sampling (LHS).
3.3.2 Acquisition Function: Condence Bound

The acquisition function is the mechanism which guides the optimization process,
implementing the trade-off between exploitation (choosing the new point in regions
which are promising according to the current knowledge) and exploration (choosing
the new point in less explored regions). This trade-off is usually achieved by com-
bining, into a single formula, both the posterior mean and posterior standard deviation,
as estimated through the surrogate model. A number of acquisition functions have been
proposed in literature, such as Probability of Improvement (PoI), Expected Improve-
ment (EI) and Condence Bound (CB). In this study the Upper Condence Bound
(UCB) is used as the goal is to maximize the clustering performance index I. The
guiding principle behind CB is to be optimistic in the face of uncertainty.
Finding the next promising point requires solving an optimization problem where
the objective function is the acquisition function. Anyway, it can be considered
inexpensive with respect to the original optimization problem. The so called focus
search algorithm, proposed in [44], is a generic approach able to deal with numeric
parameter spaces, categorical parameter spaces, as well as mixed and hierarchical
spaces. This is the procedure adopted in this study.
3.3.3 Termination Criterion

Several termination criteria are possible: the maximum number of function evaluations
is a very common choice. Other possibilities are a limit over the wall clock time, no
improvements in the best seen (i.e. the best value of objective function seen so far)
after a given number of consecutive SMBO iterations, or the difference from the
optimum when known is lower than a given threshold. Other more sophisticated
criteria have been also proposed, e.g. in [44] a Resource-Aware Model-Based Opti-
mization framework that leads to efcient utilization of parallel computer architectures
through resource-aware scheduling strategies, while in [45] the Lipschitz continuity
assumption, quite realistic for many practical black-box problems, is considered for
obtaining a global optimum estimates after performing a limited number of functions
evaluations.
The termination criteria dened in this study is the maximum number of functions
evaluations: this choice allows for an easy comparison between GP and RF based
SMBO as well as between SMBO and a simple grid search.
3.3.4 Software Environment

The implementation of the hyperparameters optimization proposed in this study has
been based on the R software environment. The recently proposed toolbox mlrMBO
[40] has been used to implement the SMBO process, along with the kernel k-means
algorithm, namely kkmeans, provided by the R package kernlab.
mlrMBO is a flexible and comprehensive R toolbox for model-based optimization
(MBO). It can deal with both single- and multi-objective optimization, as well as
continuous, categorical, mixed and conditional parameters. Additional features include
multi-point batch proposal, parallelization, visualization, logging and error-handling.
Very important, mlrMBO is implemented in a modular fashion, such that single
components can be easily replaced or adapted by the user for specic use cases.
4 Results and Discussion
This section summarizes the results obtained. The experimental setting consists in:
Grid search vs SMBO (using both GP and RF to generate the surrogate of the
objective function): with k = 3, , 13 and r in [0.00001, 0.1], with 70 values for r,
equally distributed in the range, are used for the grid search;
Maximization of the index I used to measure the performance of clustering output
with respect to leakage localization properties (as dened in Sect. 2.1):
Termination criteria, based on a limit on the number of function evaluations: 770
function evaluations (equals to the number of congurations into the grid),
where 220 are used as initial design.
The following Table 1 summarizes the results obtained:
Table 1. Results: best performance, hyperparameter conguration, time and iterations among
the three approaches (Grid Search, GP- and RF- based SMBO)
Clustering performance k* r* Time Last iteration with
I (best seen) (sec) improvement
Grid 0,516 3 0,0000100 6165,03 NA
search
GP 0,505 3 0,0055322 9704,88 388
RF 0,556 3 0,0000112 12317,92 687
The best seen of the performance index I, over the 770 function evaluations;
Values of the hyperparameters, K* and r*, associated to the best seen;
Overall execution time (sec), computed as total on all the 770 evaluations;
Last iteration with improvement of the clustering performance index I.
SMBO using RF proved to be the most effective strategy. In particular, it was able
to identify a hyperparameters conguration of the kernel k-means outside the grid
and associated to the highest value of the clustering performance index I.
On the contrary, SMBO using GP was not able to converge to a better hyperpa-
rameters conguration than the one identied by the grid search. This was probably due
to the nature of k, indeed RF is usually preferred to GP in the case of categorical
variables.
The following Fig. 3 compares the convergence of the different approaches. SMBO
with GP converges very fast after 388 evaluations of the objective function no more
improvements are obtained, even if the best seen value of I is lower than the one
obtained through the grid search.
It is important to highlight that SMBO with RF provides the following benets:
it is able to nd a hyperparameters conguration outperforming grid search as well
as SMBO with GP in terms of effectiveness (clustering performance index, I);
Fig. 3. Best seen over the iterations for SMBO with RF, SMB with GP and the best value
obtained over the grid search (independent on the iterations)
a hyperparameters conguration outperforming the grid search can be obtained after

only 338 function evaluations, well lower the limit on the number of evaluations.
The second point is really important: although the best seen after 338 evaluations is
lower than the nal one, this means that the SMBO with RF outperforms grid search
also in terms of efciency, just using about half of the maximum number of evaluations
available. Indeed, even with a more tightening termination criteria (e.g. 385 evalua-
tions, that is 770/2), without any modication to the grid search, SMBO with RF
should was able to outperform grid search, with a consequent drastic reduction of the
wall clock time, too (approximately 3370 s).
Finally, to further highlight the benets provided by SMBO for hyperparameters
conguration, we decided to give more chances to the grid search, increasing the
number of congurations to 2310 (770 3) by choosing 210 values for r, equally
distributed, in the range 0.00001 to 0.1. As result, the grid search was not able to
improve in terms of clustering performance index I even if the overall wall clock time
went from 6165,63 s to 17555,37 s.
This further result conrms the main drawback of the grid search for hyperpa-
rameters optimization: it performs many function evaluations in non-promising areas of
the search space, basically wasting computational resources.
With respect to the SVM, it has been trained according to the cluster assignment
provided by the kernel k-means instead of SC. The same SVM conguration used in
the previous study was adopted. Classication performances were similar, slightly
better in the case of the kernel k-means, conrming the equivalence between SC and
kernel k-means while enabling a signicant reduction in terms of computational
complexity in the clustering procedure.
5 Conclusions
Any performance comparison between Bayesian Optimization and other global opti-
mization strategies can be only platform and problem dependent and thus difcult to
generalize: in [19] it is stated that random search offers a simple, parallelizable and
theoretically sound launching point while Bayesian Optimization may offer improved
empirical accuracy but its selection models are intrinsically sequential and thus
difcult to parallelize. The main problem is that Bayesian Optimization scales poorly
with the number of dimensions: in [46] it is stated that the approach is restricted to
problems of moderate dimensions up to 10 dimensions and a random embedding is
proposed to identify a work space of a much smaller number of dimensions. Other
proposal to scale Bayesian Optimization to higher dimensions are in [19, 47].
The computational results reported in this paper substantiate the known fact that the
Bayesian framework is suitable to objective functions which are costly to evaluate and
black box. Moreover, they can be applied to unusual design spaces which involve
categorical or conditional inputs and are therefore able to deal with such diverse
domains as A/B testing, recommender systems, reinforcement learning, environmental
monitoring, sensor networks, preference learning and interactive interfaces.
The next activities will leverage the capability RF based SMBO to handle condi-
tional parameters as well; we will also consider the optimization of the whole machine
learning pipeline, moving towards an automatic algorithm conguration setting.
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Solution of the Convergecast Scheduling
Problem on a Square Unit Grid When
the Transmission Range is 2
Adil Erzin(B)
Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

adilerzin@math.nsc.ru
Abstract. In this paper a conict-free data aggregation problem, known

as a Convergecast Scheduling Problem, is considered. It is NP-hard in
the arbitrary wireless network. The paper deals with a special case of
the problem when the communication graph is a square grid with unit
cells and when the transmission range is 2 (in L1 metric). Earlier for
the case under consideration we proposed a polynomial time algorithm
with a guaranteed accuracy bound. In this paper we have shown that
the proposed algorithm constructs an optimal solution to the problem.
Keywords: Wireless networks Data aggregation Conict-free

scheduling
1 Introduction
In the wireless sensor networks (WSN) the data collected by the sensors should
be delivered to the analytical center. The process of transferring the packets
of information from the sensors in such a center - a base station (BS) is called
an aggregation of the data. Aggregation time (or latency), i.e. a period during
which the data from all sensors fall in the BS, is the most important criterion in
the quick response networks. The shorter aggregation time the more eectively
WSN can react to the possible events.
A synthesis of the network through which the data is transmitted, as a rule,
carried out object to the criterion of minimum communication power consump-
tion [13]. As a result, a constructed communication graph (CG) is highly sparse.
Consequently, not all vertices (sensors) can transmit the collected data directly
to the BS. Packets from the most vertices are going through the other vertices,
and the path from some vertex to the BS may consist of a big number of edges.
In the formulations of the aggregation problem the volume of the transmitted
data, as usually, does not taken into account. So the packets are considered to be
equal in length for all vertices of the CG, and each packet is transmitted along
any edge during one time round (slot).
In the majority of the wireless networks, an element (vertex or node) cannot
transmit and receive packets at the same time (half duplex mode), and a vertex
https://doi.org/10.1007/978-3-319-69404-7_4
Solution of the Convergecast Scheduling Problem 51
cannot receive more than one packet simultaneously. Moreover, due to the need of
energy saving, each sensor sends the packet once during the aggregation period
(session). This means that the packets are transmitted along the edges of a
desired aggregation tree (AT) rooted in BS, and an arbitrary vertex in the AT
must rst receive the packets from all its children, and only then can send the
aggregated packet to its parent node. So, we assume that the arcs of the AT are
oriented towards the root (BS), and if the AT is known, then the partial order
on the set of arcs of the AT is known too.
In the most WSNs, the sensors use a common radio frequency to transmit
the messages. So, if in the sensors reception area are working more than one
transmitter, then (due to the radio wave interference phenomenon), the receiver
cannot get any correct data packet. Such a situation is called a conflict or a
collision.
In the conict-free data aggregation problem it is necessary to nd an aggre-
gation tree and a conict-free schedule of minimal length [46]. This problem is
known as a Convergecast Scheduling Problem (CSP), and it is NP-hard even in
the case when the AT is given [7].
If the AT is known, it is possible to construct a graph of conicts (GC) as
follows. Each node in the GC is associated with the arc in the AT. Two vertices in
the GC are linked by an edge, if the simultaneous transmission of the respective
arcs in the AT implies a conict. There is an arc going from the node i to the
node j in the GC, if the end of the arc i coincides with the beginning of the arc
j in the AT.
It is obvious that the CSP in the case of a given AT reduces to the problem
of mixed coloring of the GC [8], which is also NP-hard, and stated as follows. Let
the mixed graph G = (V, AE) with vertex set V , the set of arcs A and the set of
edges E is given. Graph G is k-colorable if exists a function f : V {1, . . . , k}.
In the problem of mixed graph coloring it is required to nd a minimal k, for
which exists such k-coloring, that if two vertices i and j are joined by an edge
(i, j) E, then f (i) = f (j), and if there is an arc (i, j) A, then f (i) < f (j).
The problem of conict-free data aggregation has been intensively studied
by both theoreticians and practitioners [13,68,10,11]. A number of heuristic
algorithms was proposed to construct an approximate solution [46,912]. For
some of them the guaranteed accuracy bounds in terms of degree and radius
of the CG were found [13]. To assess the quality of the other heuristics the
numerical experiments were carried out [4,9].
The literature also addresses the special cases of the problem. For example,
when the conicts occur only between the children of common parent in the AT
[9]. Such a situation occurs when the sensors use dierent radio frequencies for data
transmission [12]. In this case, if AT is given, the problem can be solved in a poly-
nomial time, but when AT is a desired tree the problem remains NP-hard [14].
In [15] a special case of CG in the form of a unit square grid, in which in
each node a sensor is located, and the transmission range of each sensor is 1, is
considered. A simple polynomial algorithm for constructing an optimal solution
to this problem was proposed. In [7] a similar problem in the case when the
52 A. Erzin
transmission distance d is an arbitrary integer not less than 1 was considered. On

the one hand the increase of d can reduce the length of the schedule, but on the
other hand the number of collisions increases too. The methods of constructing
a conict-free schedule of the data transmission with the guaranteed accuracy
bounds depending on d are proposed. In particular, for the case when d = 2
was proposed an algorithm which on the (n + 1) (m + 1) grid, where n and
m are even, and the BS is in the origin (0, 0), building a schedule of length
(n + m)/2 + 3.
In this paper, we show that the lower bound for the length of the schedule
when the communication graph is a unit square grid and the transmission range
of sensors is 2 (in L1 metric), is (n + m)/2 + 3, i.e. the considered special case of
the problem is polynomially solvable, and the algorithm proposed in [7] builds
an optimal solution to the problem.
2 General Problem Formulation

Let a communication graph (CG), in which the vertices are the images of the
sensors, and two vertices are connected by an edge if a communication between
the sensors can be carried out in both directions, is given. Among the vertices
we distinguish a base station (BS) into which the data from all vertices should
be delivered. We assume that:
a time is discrete;
a transmission time of any packet along any edge of the CG is one time round;
a sensor cannot receive and transmit at the same time round;
a sensor cannot receive more than one packet during one time round;
each sensor transmits packet only once during the aggregation session;
a subset of vertices may transmit simultaneously if that does not entail a
conict;
The CSP is to nd a feasible data transmission schedule of minimum length.
To illustrate the problem, let us refer to the Fig. 1. Figure 1a shows an example
of a CG, in which BS is marked in yellow and the AT in the bold lines. If in this
CG transfers, for example, red vertex, then it is heard by 5 other vertices. Figure 1b
shows a feasible schedule, where the number inside the vertex is the time round
of transmission of this vertex. The length of the schedule in Fig. 1b is 6.
According to the CG a graph of conicts (GC) can be constructed (see.
Fig. 2). Some edges (dashed in Fig. 2b) can be deleted, since the corresponding
conicts are taken into account by the transmission order. For example, the
transmission of vertex 8 precedes the transmission of node 3, then the edge
(8, 3) can be eliminated. Figure 2c shows a feasible mixed coloring of the vertices
of the GC, in which the color number corresponds to the transfer round when
the corresponding vertex sends a packet (in Fig. 2c the number of each node is
the number of the color).
As noted above, the CSP in general and in many special cases is NP-hard.
However, in some special cases, the problem is polynomially solvable. This is so,
Fig. 1. (a) CG and AT (bold lines); (b) Feasible schedule of length 6. (Color gure
online)
Fig. 2. (a) CG and AT; (b) Graph of Conicts (GC); (c) Feasible coloring. (Color
gure online)
for example, when the CG is the unit square grid, and a transmission distance
is 1 [15]. In this paper, we are interested in the problem when a CG is also a
unit square grid, but the transmission distance is 2 (in L1 metric). In [7] for the
(n + 1) (m + 1) grid with the BS at the origin (0, 0) and when transmission
distance equals 2, an algorithm for constructing a schedule, the length of which
(when n and m are even) equals (n + m)/2 + 3, is proposed.
3 CSP in the Unit Square Grid When the Transmission

Distance is 2
Let us consider a grid graph (Fig. 3a), in which each node (x, y), x =
0, 1, . . . , n, y = 0, 1, . . . , m contains a sensor. Transmission distance of each
vertex is 2 (in L1 metric). Each sensor must send the collected data to the base
station (BS), which is located at the origin (for simplicity). The time of arrival
of the last packet in the BS is the aggregation time or the schedule length. We
denote the minimum length of the schedule as L(n, m).
54 A. Erzin
Fig. 3. (a) Grid graph; (b) Conict (infeasible) transmissions; (c) Conict-free (feasi-
ble) transmissions. (Color gure online)
The restrictions from the previous section naturally have to be met also. For
example, in Fig. 3b the unacceptable transmissions are displayed (the transmis-
sions shown by arrows of the same color cannot be performed simultaneously).
And in Fig. 3c the conict-free transmissions are indicated (the arrows of the
same color) which can be performed simultaneously.
3.1 The Exact Lower Bound for the Schedule Length

The further results are valid (with minor adjustments) for the arbitrary n and
m, but in this paper for the sake of simplicity and brevity, we set n and m to
be even.
Definition 1. The distance from the vertex to the BS is the minimum number
of transmissions from this vertex.
Then the vertex (x, y), x = 0, 1, . . . , n, y = 0, 1, . . . , m, is at a distance

](x + y)/2[, where ]a[ is the smallest integer not less than a.
Obviously, the length of the schedule cannot be less than D = (n + m)/2,
because the most remote vertex (n, m) is located at the distance D, and the
packet form it could be delivered to the BS at least during D time rounds.
Fairly obvious the next.
Property 1. If at least two vertices in the arbitrary graph are at the distance R
from the BS, then the aggregation time cannot be less than R + 1.
Since in the considered grid three vertices are at the distance D from the BS,
then follows the obvious.
Corollary 1. The aggregation time in the unit square grid (n + 1) (m + 1)

with the BS at the origin (0, 0), when the transmission distance is 2, is at least
D + 1.
Based on the Property 1 it is also easy to prove.
Corollary 2. The aggregation time in the unit square grid (n + 1) (m + 1)

with the BS at the origin (0, 0), when the transmission distance is 2, is at least
D + 2.
The main purpose of this paper is to prove the following
Lemma 1. The aggregation time in the unit square grid (n + 1) (m + 1) with

the BS at the origin (0, 0), when the transmission distance is 2, is at least D + 3.
Proof. We rst consider the dierent (up to symmetry) transmissions from the
vertex (n, m) at the moment (time round) 1 (Fig. 4). At that, let us encode each
possible action (transmission) as a (t = a; b), where a is a time round, and b
is the number/code of the allowable transmission. For example, when (t = 1; 0)
in Fig. 4 the vertex (n, m) is silent (dont send a packet), and if (t = 1; 1) it
transfers the packet to another blue vertex which is also located at the distance
D from the BS (the origin).
In the gures the node with the red circle cannot transmit a packet during
the moments 1, 2, . . . , t, but vertex with the green circle can transmit at the time
round t. For example, in the case (t = 1; 1) the receiver may hear 6 extra vertices
besides the sender. Thus, these vertices (5 yellow and one blue) must be silent
and in Fig. 4 they have the red circles.
Fig. 4. Possible transmissions from the vertex (n, m) at the moment 1. (Color gure
online)
When we proved the statement, we have considered all possible cases (more
than 100 000), but they, of course, cannot be presented all in a single paper, so
here we will describe only a few of them as the examples. So, we present not a
complete proof, but only illustrate the proof. The analysis of all cases is planned
to place in Internet in the foreseeable future.
Case (t = 1;0). Let us consider the possible actions of the vertex (n1, m) at the
time round 1 (Fig. 5), and then consider the case (t = 1; 0.0.0) in detail (when
all blue nodes (at the distance D) are silent (Fig. 6) at the moment 1).
The possible actions of the vertex (n, m) at the time round 2 are shown in
Fig. 7.
Consider the possible actions of the vertex (n, m) at the time 2 (Fig. 7), and
the detailed case (t = 2; 0) when the node (n, m) is silent. Consider the behavior
56 A. Erzin
Fig. 5. Possible transmissions from the node (n 1, m) at the moment 1, when vertex
(n, m) is silent.
Fig. 6. Possible transmissions from the node (n, m 1) at the moment 1, when vertices
(n, m) and (n 1, m) are silent. (Color gure online)
Fig. 7. Possible transmissions from the node (n, m) at the moment 2, when all blue
nodes were silent at moment 1. (Color gure online)
of the vertex (n 1, m) at the time round 2. If it is silent, then after two time
rounds remain at least two vertices at the distance D, which have not started
transmitting. Therefore, by Property 1, the length of the schedule cannot be less
than 2 + D + 1 = D + 3 (Fig. 8).
Fig. 8. Possible transmissions from the node (n 1, m) at the moment 2, when all blue
nodes were silent at moment 1, and vertex (n, m) is silent at moment 2. (Color gure
online)
Let us consider the case (t = 2; 0.1). Since the vertex (n, m) is silent, then,
according to Property 1, both vertices (n 1, m) and (n, m 1) must transmit.
The transmission cases are shown in Fig. 9.
Fig. 9. Possible transmissions from the node (n, m 1) at the moment 2 in case (t =
2; 0.1).
Further, as an example, we consider in detail the case (t = 2; 0.1.1). At

the moment 3 the vertex (n, m) must transmit a packet, else the length of the
schedule will be at least D + 3. The possible transmissions are shown in Fig. 10.
In the case (t = 3; 2) after the time round 3 remain at least 3 yellow vertices
(at the distance D 1) which has not transmitted, so the schedule length is at
least 3 + D 1 + 1 = D + 3. The case (t = 3; 1) we will examine in detail. If after
the 3rd time round remain at least 2 yellow vertices (at the distance D 1), then
the length of the schedule cannot be less than D + 3. To avoid this the circled
vertices (Fig. 11) in the case (t = 1; 0.0.0) must send the packets and this can
be done at the time round 1 in the manner shown in Fig. 11 by blue arrows. But
then the other yellow vertices, which are outlined in the case (t = 3; 1) will not
be able transmitting before the 3rd time round. However, at the moment 3 they
all cannot transmit the packets. As a result, after the 3rd time round at least
two nodes at the distance D 1 will not start transmission and therefore the
schedule length is not less than D + 3.
58 A. Erzin
Fig. 10. Possible transmissions from the node (n, m) at the moment 3 in case (t =
2; 0.1.1).
Fig. 11. Transmissions in the cases (t = 1; 0.0.0), (t = 2; 0.1.1), (t = 3; 1). (Color

gure online)
Let us consider one more case.

Case (t = 1;3). In this case, at the rst time round the vertices (n 1, m) and
(n, m 1) cannot transmit. The possible transmissions from the node (n 1, m)
at the moment 2 are shown in Fig. 12.
Next, consider, for example, the case (t = 2; 1.1) in Fig. 13. The possible
transmissions from the node (n 1, m 1) at the time round 3 are shown in
Fig. 14.
The cases (t = 3; 1) (t = 3; 7) are simple, since after the 3rd time round
remain at least two vertices at the distance D 1, which have not yet broadcast
and Property 1 implies that the length of the schedule is not less than D + 3.
The case (t = 3; 0), when the node (n 1, m 1) is silent, is dicult. Let us
consider it. That after the 3rd moment remains not more than one silent yellow
vertex (at a distance D 1), it is necessary to send the packets from all other
yellow vertices (at the distance D 1), except the vertex (n 1, m 1), not
later than at the time round 3. From the vertices (n 2, m), (n 2, m 1),
(n 1, m 2) and (n, m 2) the packets can be sent only at the moment 3, and
the only way to do so is shown in Fig. 15 (t = 3; 0) with the blue arrows. Then
from the vertices (n 3, m) and (n, m 3) the packets can be sent only at the
moment 1, for example, as it is shown in Fig. 15 (t = 1; 3) with the blue arrows.
Let us consider the moment 4 and the variants of sending a packet from the
vertex (n 1, m 1). If this node will not send the packet at the moment 4, the
schedule length cannot be less than D + 3. The possible transmissions are shown
in Fig. 16. As a result, after the 4th time round at least two green nodes (at the
distance D 2) remain silent. So (by Property 1), we have that the length of the
schedule is at least 4 + D 2 + 1 = D + 3.
Fig. 12. Possible transmissions from the node (n 1, m) at the moment 2.
Fig. 13. Possible transmissions from the node (n, m 1) in the case (t = 2; 1).
Fig. 14. Possible transmissions from the node (n 1, m 1) at the moment 3 in the
case (t = 2; 1.1).
The author have considered all possible transmission from the vertices at a
distance of at least D 2 from the BS (the origin), and it is shown that the
length of the schedule cannot be less than D + 3, which proves the lemma.
60 A. Erzin
Fig. 15. Cases (t = 1; 3), (t = 2; 1.1) and (t = 3; 0). (Color gure online)
Fig. 16. Possible transmissions from the node (n 1, m 1) at the moment 4. (Color
gure online)
3.2 Algorithm A
In this section, we present a version of the pseudo-code of the algorithm A which
is proposed in [7]. The set of vertices with identical ordinates for convenience we
call a layer.
Algorithm A.
Step 1. Set t = 1.
Send from all even vertices (0, m 1), (2, m 1), . . . , (n, m 1) at the layer
m 1 the packets up to a distance 1 to the corresponding vertices at the layer
m.
Send from all vertices at the layer m 3 the packets down to a distance of
2, i.e., to the corresponding nodes at the layer m 5.
Send from all even vertices (0, 1), (2, 1), . . . , (n, 1) at the layer 1 the packets
up to a distance 1 to the corresponding vertices at the layer 2.
Step 2. Set t = 2.
Send from all odd vertices (1, m 1), (3, m 1), . . . , (n 1, m 1) at the
layer m 1 the packets up to a distance of 1 to the corresponding vertices at
the layer m.
Send from all vertices at the layer m 5 the packets down to a distance of
2, i.e. to the corresponding vertices at the layer m 7.
Send from all odd vertices (1, 1), (3, 1), . . . , (n 1, 1) at the layer 1 the
packets up to a distance 1 to the corresponding nodes at the layer 2.
Step 3. Set t = t + 1 and k = m 2(t 3).
Send from all vertices at the layer k the packets down to a distance 2 to the
corresponding vertices at the layer k 2.
Send from all vertices at the layer k 2t 1 the packets down to a distance
of 2 to the corresponding nodes at the layer k 2t 3.
If k 2t 1 > 3, then go to Step 3.

Send from all vertices at the layer k the packets down at distance 2 to the
Send from all even vertices (0, 3), (2, 3), . . . , (n, 3) at the layer 3 the packets
down to a distance 1 to the corresponding vertices at the layer 2.
Send from all odd vertices (1, 3), (3, 3), . . . , (n1, 3) at the layer 3 the packets
down to a distance 2 to the corresponding nodes at the layer 2.
If k > 2, then go to Step 6.
After t = m/2 + 2 time rounds we have a linear graph with n + 1 vertices
(vertex 0 is a BS), which consists of the set of vertices with the coordinates
(x, 0), x = 0, 1, . . . , n, and wherein the distance between the adjacent vertices is
1. We enumerate the vertices of this graph, starting from the BS, by the natural
numbers 0, 1, . . . , n.
Step 7. Set t = t + 1.
Send the packets from the odd vertices n 1, n 7, . . . , n 6k 1, . . . , 1, k
(n 2)/6, at the distance 1 to the right, and the packets from the odd vertices
n 3, n 9, . . . , n 6k 4, . . . , 3, k (n 7)/6, send left at a distance of 1.
Step 8. Set t = t + 1.
Send the packets from the odd vertices that were silent during the previous
steps, to the left at a distance of 1.
Send a packet from the vertex n to the vertex n 2.
Set k = n.
Step 9. Set t = t + 1 and k = k 2.
Send a packet from the vertex k to the vertex k 2.
If k > 2, then go to Step 9.
Stop.
Note that the vertical aggregation is carried out with time complexity O(m),
and the horizontal aggregation is carried out with time complexity O(n). There-
fore, the complexity of algorithm A is O(n + m).
Algorithm A returns a schedule which length is D + 3. From Lemma 1 we
know that the aggregation time cannot be less than D + 3. Hence, the following
theorem holds.
Theorem 1. Algorithm A builds an optimal conflict-free schedule of data aggre-

gation in the unit square grid (n + 1) (m + 1), with even n and m, with the
BS at the origin (0, 0) and a transmission distance equals 2, the length of which
is (n + m)/2 + 3.
62 A. Erzin
4 Conclusion
In this paper we found an exact lower bound for the length of the conict-free
schedule of data aggregation in the unit square grid (unit disk graph in L1 metric)
when the transmission range of each vertex is 2. This lower bound coincides with
the length of the schedule constructed by algorithm A in a polynomial time.
Consequently, polynomial time algorithm A constructs an optimal schedule, the
length of which is L(n, m) = (n + m)/2 + 3.
Acknowledgments. This work was supported by RFBR (grant 16-07-00552) and

the Ministry of Education and Science of the Republic of Kazakhstan (project No.
0115PK00550).
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1007/978-3-662-46018-4 10
A GRASP for the Minimum Cost SAT Problem
Giovanni Felici1 , Daniele Ferone2 , Paola Festa2(B) , Antonio Napoletano2 ,

and Tommaso Pastore2
1
Institute for Systems Analysis and Computer Science,
IASI-CNR, 00185 Rome, Italy
giovanni.felici@iasi.cnr.it
2
Department of Mathematics and Applications, University of Napoli Federico II,
Compl. MSA, Via Cintia, 80126 Napoli, Italy
{daniele.ferone,paola.festa,antonio.napoletano2,tommaso.pastore}@unina.it
Abstract. A substantial connection exists between supervised learn-

ing from data represented in logic form and the solution of the Mini-
mum Cost Satisability Problem (MinCostSAT). Methods based on such
connection have been developed and successfully applied in many con-
texts. The deployment of such methods to large-scale learning problem is
often hindered by the computational challenge of solving MinCostSAT,
a problem well known to be NP-complete. In this paper, we propose
a GRASP-based metaheuristic designed for such problem, that proves
successful in leveraging the very distinctive structure of the MinCost-
SAT problems arising in supervised learning. The algorithm is equipped
with an original stopping criterion based on probabilistic assumptions
which results very eective for deciding when the search space has been
explored enough. Although the proposed solver may approach MinCost-
SAT of general form, in this paper we limit our analysis to some instances
that have been created from articial supervised learning problems, and
show that our method outperforms more general purpose well established
solvers.
Keywords: Hard combinatorial optimization SAT problems

GRASP Local search Probabilistic stopping criterion Approximate
solutions
1 Introduction
Propositional Satisability (SAT) and its derivations are well known problems
in logic and optimization, and belong to the special class of NP-complete prob-
lems [11]. Beside playing a special role in the theory of complexity, they often
arise in applications, where they are used to model complex problems whose
solution is of particular interest.
One such case surfaces in logic supervised learning. Here, we have a dataset of
samples, each represented by a nite number of logic variables, and a particular
extension of the classic SAT problem - the Minimum Cost Satisability Problem
https://doi.org/10.1007/978-3-319-69404-7_5
A GRASP for the Minimum Cost SAT Problem 65
(MinCostSAT) - can be used to iteratively identify the dierent clauses of a

compact formula in Disjunctive Normal Form (DNF) that possesses the desirable
property of assuming the value True on one specic subset of the dataset and
the value False on the rest.
The use of MinCostSAT for learning propositional formula from data is
described in [6,25], and its description is beyond the scope of this paper. Suce
it to say, however, that there are several reasons that motivate the validity of
such an approach to supervised learning, and that it proved to be very eective
in several applications, particularly on those derived from biological and medical
data analysis [1,3,4,2729].
One of the main drawbacks of the approach described in [6] lies in the di-
culty of solving MinCostSAT exactly or with an appropriate quality level. Such
drawback is becoming more and more evident as, in the era of Big Data, the
size of the datasets that one is to analyze steadily increases. Feature selection
techniques may be used to reduce the space in which the samples are represented
(one such method specically designed for data in logic form is described in [2]).
While the literature proposes both exact approaches ([10,20,25,26]) and
heuristics ([23]), still the need for ecient MinCostSAT solvers remains, and
in particular for solvers that may take advantage of the specic structure of
those MinCostSAT representing supervised learning problems.
In this paper, we try to ll this gap and propose a GRASP-based metaheuris-
tic designed to solve MinCostSAT problems that arise in supervised learning. In
doing so, we developed a new probabilistic stopping criterion that proves to be
very eective in limiting the exploration of the solution space - whose explosion
is a frequent problem in metaheuristic approaches. The method has been tested
on several instances derived from articial supervised problems in logic form,
and successfully compared with four established solvers in the literature (Z3
from Microsoft Research [19], bsolo [12], MiniSat+ [5], and PWBO [1517]).
The paper is organized as follows. Section 2 presents a very simple formula-
tion of the problem; Sect. 3 describes the architecture of the GRASP algorithm;
Sect. 4 is devoted to the explanation of the probabilistic stopping criterion, while
experimental results and conclusions are covered in Sects. 5 and 6, respectively.
2 Mathematical Formulation of the Problem

The Minimum Cost SAT Problem (MinCostSAT) - also known as Binate Cover-
ing Problem - is a special case of the well known Boolean Satisability Problem.
Given a set of n boolean variables X = {x1 , . . . , xn }, a non-negative cost func-
tion c : X R+ such that c(xi ) = ci 0, i = 1, . . . , n, and a boolean formula
(X) expressed in CNF, the MinCostSAT problem consists in nding a truth
assignment for the variables in X such that the total cost is minimized while
66 G. Felici et al.
(X) is satised. Accordingly, the mathematical formulation of the problem is

given as follows:
n

(MinCostSAT) z = min ci xi
i=1
subject to:
(X) = 1,
xi {0, 1}, i = 1, . . . , n.
It is easy to see that a general SAT problem can be reduced to a MinCostSAT

problem whose costs ci are all equal to 0. Furthermore, the decision version of
the MinCostSAT problem is NP-complete [10]. While the boolean satisability
problem is an evergreen in the landscape of scientic literature, MinCostSAT
has received less attention.
3 A GRASP for MinCostSAT

GRASP is a well established iterative multistart metaheuristic method for di-
cult combinatorial optimization problems [7]. The reader can refer to [8,9] for a
study of a generic GRASP metaheuristic framework and its applications.
Such method is characterized by the repeated execution of two main phases:
a construction and a local search phase. The construction phase iteratively adds
one component at a time to the current solution under construction. At each iter-
ation, an element is randomly selected from a restricted candidate list (RCL),
composed by the best candidates, according to some greedy function that mea-
sures the myopic benet of selecting each element.
Once a complete solution is obtained, the local search procedure attempts
to improve it by producing a locally optimal solution with respect to some suit-
ably dened neighborhood structure. Construction and local search phases are
repeatedly applied. The best locally optimal solution found is returned as nal
result. Figure 1 depicts the pseudo-code of a generic GRASP for a minimization
problem.
In order to allow a better and easier implementation of our GRASP, we
treat the MinCostSAT as particular covering problem with incompatibility con-
straints. Indeed, we consider each literal (x, x) as a separate element, and a
clause is covered if at least one literal in the clause is contained in the solution.
The algorithm tries to add literals to the solution in order to cover all the clauses
and, once the literal x is added to the solution, then the literal x cannot be
inserted (and vice versa). Therefore, if the literal x is in solution, the variable
x is assigned to true and all clauses covered by x are satised. Similarly, if the
literal x is in solution, the variable x is assigned to false, and clauses containing
x are satised.
The construction phase adds a literal at a time, until all clauses are covered
or no more literals can be assigned. At each iteration of the construction, if a
1 Algorithm GRASP()
2 x Nil ;
3 z(x ) + ;
4 while a stopping criterion is not satisfied do
5 Build a greedy randomized solution x ;
6 x LocalSearch(x) ;
7 if z(x) < z(x ) then
8 x x ;
9 z(x ) z(x) ;
10 return x
Fig. 1. A generic GRASP for a minimization problem.
clause can be covered only by a single literal x due to the choices made in
previous iterations then x is selected to cover the clause. Otherwise, if there
are not clauses covered by only a single literal, the addition of literals to the
solution takes place according to a penalty function penalty(), which greedily
sorts all the candidates literals, as described below.
Let cr(x) be the number of clauses yet to be covered that contain x. We then
compute:
c(x) + cr(x)
penalty(x) = . (1)
cr(x)
This penalty function evaluates both the benets and disadvantages that
can result from the choice of a literal rather than another. The benets are
proportional to the number of uncovered clauses that the chosen literal could
cover, while the disadvantages are related to both the cost of the literal and
the number of uncovered clauses that could be covered by x. The smaller the
penalty function penalty(x), the more favorable is the literal x. According to the
GRASP scheme, the selection of the literal to add is not purely greedy, but a
Restricted Candidate List (RCL) is created with the most promising elements,
and an element is randomly selected among them. Concerning the tuning of the
parameter , whose task is to adjust the greediness of the construction phase,
we performed an extensive analysis over a set of ten dierent random seeds.
Such testing showed how a nearly totally greedy setup ( = 0.1) allowed the
algorithm to attain better quality solutions in smallest running times.
Let |C| = m be the number of clauses. Since |X| = 2n, in the worst case
scenario the loop while (Fig. 2, line 3) in the construct-solution function
pseudo-coded in Fig. 2 runs m times and in each run the most expensive opera-
tion consists in the construction of the RCL. Therefore, the total computational
complexity is O(m n).
In the local search phase, the algorithm uses a 1-exchange (ip) neighborhood
function, where two solutions are neighbors if and only if they dier in at most one
component. Therefore, if there exists a better solution x that diers only for one
literal from the current solution x, the current solution s is set to s and the proce-
dure restarts. If such a solution does not exists, the procedure ends and returns the
68 G. Felici et al.
1 Function construct-solution(C, X, )
/* C is the set of uncovered clauses */
/* X is the set of candidate literals */
2 s;
3 while C = do
4 if c C can be covered only by x X then
5 s s {x};
6 X X \ {x, x};
7 C C \ {c | x c};
8 else
9 compute penalty(x) x X;
10 th min{penalty(x)} + (max{penalty(x)} min{penalty(x)}) ;
xX xX xX
11 RCL { x X : penalty(x) th } ;
12 x rand(RCL) ;
13 s s {x};
14 X X \ {x, x};
15 C C \ {c | x c};
16 return s
Fig. 2. Pseudo-code of the GRASP construction phase.
current solution s. The local search procedure would also re-establish feasibility
if the current solution is not covering all clauses of (X). During our experimen-
tation we tested the one-ip local search using two dierent neighborhood explo-
ration strategies: rst improvement and best improvement. With the former strat-
egy, the current solution is replaced by the rst improving solution found in its
neighborhood; such improving solution is then used as a starting point for the next
local exploration. On the other hand, with the best improvement strategy, the cur-
rent solution x is replaced with the solution x N (x) corresponding to the greatest
improvement in terms of objective function value; x is then used as a starting point
for the next local exploration. Our results showed how the rst improvement strat-
egy is slightly faster, as expected, while attaining solution of the same quality of
those given by the best improvement strategy. Based on this rationale, we selected
rst improvement as exploration strategy in our testing phase.
4 Probabilistic Stopping Rule

Although being very fast and eective, most metaheuristics present a shortcom-
ing in the eectiveness of their stopping rule. Usually, the stopping criterion
is based on a bound on the maximum number of iterations, a limit on total
execution time, or a given maximum number of consecutive iterations with-
out improvement. In this paper, we propose a probabilistic stopping criterion,
inspired by [21].
The stopping criterion is composed of two phases, described in the next
subsections. It can be sketched as follows. First, let X be a random variable
representing the value of a solution obtained at the end of a generic GRASP

iteration. In the rst phase the fitting-data procedure the probabil-
ity distribution fX () of X is estimated, while during the second phase
improve-probability procedure the probability of obtaining an improvement
of the current solution value is computed. Then, accordingly to a threshold, the
algorithm either stops or continues its execution.
4.1 Fitting Data Procedure
The rst step to be performed in order to properly represent the random vari-
able X with a theoretical distribution consists in an empirical observation of
the algorithm. Examining the objective function values obtained at the end
of each iteration, and counting up the respective frequencies, it is possible to
select a promising parametric family of distributions. Afterwards, by means of
a Maximum Likelihood Estimation (MLE), see for example [22], a choice is made
regarding the parameters characterizing the best tting distribution of the cho-
sen family.
In order to carry on the empirical analysis of the objective function value
obtained in a generic iteration of GRASP, which will result in a rst guess
concerning the parametric family of distributions, we represent the data obtained
in the following way.
Let I be a xed instance and F the set of solutions obtained by the algo-
rithm up to the current iteration, and let Z be the multiset of the objective
function values associated to F. Since we are dealing with a minimum optimiza-
tion problem, it is harder to nd good quality solutions, whose cost is small in
term of objective function, rather than expensive ones. This means that during
the analysis of the values in Z we expect to nd an higher concentration of
elements between the mean value and the max(Z). In order to represent the
values in Z with a positive distribution function, that presents higher frequencies
in a right neighborhood of zero and a single tail which decays for growing values
of the random variable, we perform a reection on the data in Z by means the
following transformation:
z = max(Z) z, z Z. (2)
The behaviour of the distribution of z in our instances has then a very recog-
nizable behaviour. A representative of such distribution is given in Fig. 3 where
the histogram of absolute and relative frequencies of z are plotted. It is easy
to observe how the gamma distribution family represents a reasonable educated
guess for our random variable.
Once we have chosen the gamma distribution family, we estimate its parame-
ters performing a MLE. In order to accomplish the estimation, we collect an initial
sample of solution values and on-line execute a function, developed in R (whose
pseudo-code is reported in Fig. 4), which carries out the MLE and returns the
characteristic shape and scale parameters, k and , which pinpoint the specic
distribution of the gamma family that best suits the data.
70 G. Felici et al.
Fig. 3. Empirical analysis of frequencies of the solutions.
1 Function fitting-data(Z)
/* Z is the initial sample of the objective function values */
2 foreach z Z do
3 z = max(Z) z;
4 {k, } MLE(Z, gamma);
5 return {k, }
Fig. 4. Fitting data procedure.
4.2 Improve Probability Procedure

The second phase of the probabilistic stop takes place once that the probability
distribution function of the random variable X , fX () has been estimated.
Let z be the best solution value found so far. It is possible to compute an
approximation of the probability of improving the incumbent solution by
max(Z) z
p=1 fX (t) dt. (3)
0
The result of the procedures fitting-data and improve-probability con-
sists in an estimate of the probability of incurring in an improving solution in
the next iterations. Such probability is compared with a user-dened threshold,
, and if p < the algorithm stops. More specically, in our implementation
the stopping criterion works as follows:
(a) let q be an user-dened positive integer, and let Z be the sample of initial
solution values obtained by the GRASP in the rst q iterations;
(b) call the fitting-data procedure, whose input is Z is called one-o to esti-
mate shape and scale parameters, k and , of the best tting gamma distri-
bution;
(c) every time that an incumbent is improved, improve-probability proce-
dure (pseudo-code in Fig. 5) is performed and the probability p of further
improvements is computed. If p is less than or equal to the stopping crite-
rion is satised. For the purpose of determining p, we have used the function
pgamma of R package stats.
1 Function improve-probability(k, , z )
/* z is the value of the incumbent */
2 p pgamma(z , shape = k, scale = );
3 return p
Fig. 5. Improve probability procedure.
5 Results
Our GRASP has been implemented in C++ and compiled with gcc 5.4.0 with
the ag -std=c++14. All tests were run on a cluster of nodes, connected by
10 Gigabit Inniband technology, each of them with two processors Intel Xeon
E5-4610v2@2.30 GHz.
We performed two dierent kinds of experimental tests. In the rst one, we
compared the algorithm with dierent solvers proposed in literature, without
use of probabilistic stop. In particular, we used: Z3 solver freely available from
Microsoft Research [19], bsolo solver kindly provided by its authors [12], the
MiniSat+ [5] available at web page http://minisat.se/, and PWBO available
at web page http://sat.inesc-id.pt/pwbo/index.html. The aim of this rst set
of computational experiment is the evaluation of the quality of the solutions
obtained by our algorithm within a certain time limit. More specically, the
stopping criterion for GRASP, bsolo and PWBO is a time limit of 3 h, for Z3
and MiniSat+ is the reaching of an optimal solution.
Z3 is a satisability modulo theories (SMT) solver from Microsoft Research
that generalizes boolean satisability by adding equality reasoning, arithmetic,
xed-size bit-vectors, arrays, quantiers, and other useful rst-order theories.
Z3 integrates modern backtracking-based search algorithm for solving the CNF-
SAT problem, namely DPLL-algorithm; in addition it provides a standard search
pruning methods, such as two-watching literals, lemma learning using conict
clauses, phase caching for guiding case splits, and performs non-chronological
backtracking.
bsolo [12,13] is an algorithmic scheme resulting from the integration of sev-
eral features from SAT-algorithms in a branch-and-bound procedure to solve the
binate covering problem. It incorporates the most important characteristics of a
branch-and-bound and SAT algorithm, bounding and reduction techniques for
the former, and search pruning techniques for the latter. In particular, it incor-
porates the search pruning techniques of the Generic seaRch Algorithm-SAT
proposed in [14].
MiniSat+ [5,24] is a minimalistic implementation of a Cha-like SAT solver
based on the two-literal watch scheme for fast boolean constraint propagation
[18], and conict clauses driven learning [14]. In fact the MiniSat solver provides
a mechanism which allows to minimize the clauses conicts.
PWBO [1517] is a Parallel Weighted Boolean Optimization Solver. The
algorithm uses two threads in order to simultaneously estimate a lower and an
upper bound, by means of an unsatisability-based procedure and a linear search,
72 G. Felici et al.
respectively. Moreover, learned clauses are shared between threads during the
search.
For testing, we have initially considered the datasets used to test feature
selection methods in [2], where an extensive description of the generation pro-
cedure can be found. Such testbed is composed of 4 types of problems (A,B,C,D),
for each of which 10 random repetitions have been generated. Problems of type
A and B are of moderate size (100 positive examples, 100 negative examples, 100
logic features), but dier in the form of the formula used to classify the samples
into the positive and negative classes (the formula being more complex for B
than for A). Problems of type C and D are much larger (200 positive exam-
ples, 200 negative examples, 2500 logic features), and D has a more complex
generating logic formula than C.
Table 1 reports both the value of the solutions and the time needed to achieve
them (in the case of GRASP, it is average over ten runs).1 For problems of
moderate size (A and B), the results show that GRASP nds an optimal solution
whenever one of the exact solvers converges. Moreover, GRASP is very fast in
nding the optimal solution, although here it runs the full allotted time before
stopping the search. For larger instances (C and D), GRASP always provides a
solution within the bounds, while two of the other tested solvers fail in doing so
and the two that are successful (bsolo, PWBO) always obtain values of inferior
quality.
The second set of experimental tests was performed for the purpose of evaluat-
ing the impact of the probabilistic stopping rule. In order to do so, we have chosen
ve dierent values for threshold , two distinct sizes for the set Z of initial solu-
tion, and executed GRASP using ten dierent random seeds imposing a maximum
number of iterations as stopping criterion. This experimental setup yielded for each
instance, and for each threshold value, 20 executions of the algorithm. About such
runs, the data collected were: the number of executions in which the probabilis-
tic stopping rule was veried (stops), the mean value of the objective function
of the best solution found (z ), and the average computational time needed (t ).
To carry out the evaluation of the stopping rule, we executed the algorithm only
using the maximum number of iterations as stopping criterion for each instance
and for each random seed. About this second setup, the data collected are, as for
the rst one, the objective function of the best solution found (z ) and the aver-
age computational time needed (t ). For the sake of comparison, we considered the
percentage gaps between the results collected with and without the probabilistic
stopping rule. The second set of experimental tests is summarized in Table 2 and
in Fig. 7. For each pair of columns (3,4), (6,7), (9,10), (12, 13), the table reports the
percentage of loss in terms of objective function value and the percentage of gain
in terms of computation times using the probabilistic stopping criterion, respec-
tively. The analysis of the gaps shows how the probabilistic stop yields little or no
changes in the objective function value while bringing dramatic improvements in
the total computational time.
1
For missing values, the algorithm was not able to nd the optimal solution in 24 h.
Table 1. Comparison between GRASP and other solvers.
GRASP Z3 bsolo MiniSat+ pwbo-2T

Inst. Time Value Time Value Time Value Time Value Time Value
A1 6.56 78.0 10767.75 78.0 0.09 78.0 0.19 78.0 0.03 78.0
A2 1.71 71.0 611.29 71.0 109.59 71.0 75.46 71.0 121.58 71.0
A3 0.64 65.0 49.75 65.0 598.71 65.0 10.22 65.0 5.14 65.0
A4 0.18 58.0 4.00 58.0 205.77 58.0 137.82 58.0 56.64 58.0
A5 0.29 66.0 69.31 66.0 331.51 66.0 9.03 66.0 30.64 66.0
A6 21.97 77.0 5500.17 77.0 328.93 77.0 32.82 77.0 359.97 77.0
A7 0.21 63.0 30.57 63.0 134.20 63.0 19.34 63.0 24.12 63.0
A8 0.25 62.0 6.57 62.0 307.69 62.0 16.84 62.0 11.81 62.0
A9 12.79 72.0 1088.83 72.0 3118.32 72.0 288.76 72.0 208.63 72.0
A10 0.33 66.0 42.23 66.0 62.03 66.0 37.75 66.0 1.81 66.0
B1 6.17 78.0 8600.60 78.0 304.36 78.0 121.25 78.0 20.01 78.0
B2 493.56 80.0 18789.20 80.0 4107.41 80.0 48.21 80.0 823.66 80.0
B3 205.37 77.0 7037.00 77.0 515.25 77.0 132.74 77.0 1.69 77.0
B4 38.26 77.0 7762.03 77.0 376.00 77.0 119.49 77.0 1462.18 77.0
B5 19.89 79.0 15785.35 79.0 3025.26 79.0 214.52 79.0 45.05 79.0
B6 28.45 76.0 4087.14 76.0 394.45 76.0 162.31 76.0 83.72 76.0
B7 129.76 78.0 10114.84 78.0 490.30 78.0 266.25 78.0 455.92 81.0
B8 44.42 76.0 5186.45 76.0 5821.19 76.0 1319.21 76.0 259.07 76.0
B9 152.77 80.0 14802.00 80.0 5216.95 82.0 36.28 80.0 557.02 80.0
B10 7.55 73.0 1632.87 73.0 760.28 79.0 370.30 73.0 72.09 73.0
C1 366.24 132.0 86400 8616.25 178.0* 86400 343.38 178.0
C2 543.11 131.0 86400 323.90 150.0* 86400 1742.68 174.0
C3 5883.6 174.1 86400 6166.06 177.0* 86400 421.64 177.0
C4 4507.63 176.3 86400 6209.69 178.0* 86400 2443.20 177.0
C5 5707.51 171.2 86400 314.18 179.0* 86400 67.73 178.0
C6 6269.91 172.1 86400 1547.90 177.0* 86400 2188.82 177.0
C7 6193.15 165.9 86400 794.90 177.0* 86400 730.36 178.0
C8 596.58 137.0 86400 306.27 169.0* 86400 837.71 178.0
C9 466.3 136.0 86400 433.32 179.0* 86400 3455.92 178.0
C10 938.54 136.0 86400 3703.94 180.0* 86400 4617.24 179.0
D1 3801.61 145.3 86400 307.25 175.0* 86400 127.69 180.0
D2 2040.64 139.0 86400 7704.92 177.0* 86400 2327.23 177.0
D3 1742.78 143.0 86400 309.10 145.0* 86400 345.97 178.0
D4 1741.95 135.0 86400 6457.79 177.0* 86400 295.76 178.0
D5 1506.22 134.0 86400 6283.27 178.0* 86400 238.81 173.0
D6 1960.87 144.5 86400 309.11 173.0* 86400 2413.42 178.0
D7 1544.42 143.0 86400 4378.73 179.0* 86400 1250.07 178.0
D8 1756.15 144.0 86400 1214.97 179.0* 86400 248.85 179.0
D9 2779.38 137.0 86400 303.11 146.0* 86400 4.73 179.0
D10 5896.86 149.0 86400 319.45 170.0* 86400 1239.93 176.0
Y 16.05 0.0 0.73 0.0 9411.06 974* 1.96 0 0.23 0.0
*sub-optimal solution
no optimal solution found in 24 h
74 G. Felici et al.
Table 2. Probabilistic stop on instances A, B, C and D.
threshold inst %-gap z %-gap t(s) inst %-gap z %-gap t(s) inst %-gap z %-gap t(s) inst %gap z %gap t(s)
5 102 A1 -0.0 83.1 B1 -2.1 87.1 C1 -6.6 76.0 D1 -5.0 79.3
1 102 A1 -0.0 83.1 B1 -2.1 87.1 C1 -6.6 76.1 D1 -5.0 79.3
5 103 A1 -0.0 83.0 B1 -2.1 87.1 C1 -5.0 74.8 D1 -4.9 78.7
1 103 A1 -0.0 2.5 B1 -2.1 87.1 C1 -3.8 70.7 D1 -1.7 58.9
5 104 A1 -0.0 -15.3 B1 -2.1 87.2 C1 -2.6 70.2 D1 -1.2 49.0
1 104 A1 -0.0 -11.8 B1 -0.5 86.1 C1 -1.3 52.5 D1 -0.2 31.6
5 102 A2 -0.0 84.0 B2 -0.7 87.0 C2 -3.5 76.0 D2 -0.1 79.1
1 102 A2 -0.0 84.1 B2 -0.7 87.0 C2 -3.5 76.2 D2 -0.1 79.1
5 103 A2 -0.0 83.6 B2 -0.7 86.9 C2 -3.5 76.7 D2 -0.1 79.1
1 103 A2 -0.0 84.0 B2 -0.7 87.0 C2 -1.9 76.4 D2 -0.1 79.1
5 104 A2 -0.0 84.9 B2 -0.7 87.0 C2 -1.9 76.1 D2 -0.1 75.7
1 104 A2 -0.0 57.9 B2 -0.1 71.3 C2 -1.9 65.2 D2 -0.1 53.5
5 102 A3 -0.0 83.4 B3 -2.7 87.0 C3 -2.7 76.3 D3 -1.8 75.2
1 102 A3 -0.0 83.8 B3 -2.7 87.0 C3 -2.1 73.0 D3 -1.8 75.2
5 103 A3 -0.0 82.9 B3 -2.7 87.0 C3 -1.7 68.0 D3 -1.7 74.8
1 103 A3 -0.0 8.3 B3 -2.6 86.6 C3 -0.6 40.9 D3 -0.8 38.5
5 104 A3 -0.0 -1.6 B3 -2.0 84.1 C3 -0.0 28.3 D3 -0.5 19.1
1 104 A3 -0.0 -6.8 B3 -0.7 58.4 C3 -0.0 9.9 D3 -0.3 14.5
5 102 A4 -0.0 86.4 B4 -2.3 86.9 C4 -4.3 78.8 D4 -2.2 75.0
1 102 A4 -0.0 6.4 B4 -2.3 86.9 C4 -3.3 68.0 D4 -2.2 70.9
5 103 A4 -0.0 3.5 B4 -2.3 86.9 C4 -2.2 63.9 D4 -2.2 66.8
1 103 A4 -0.0 1.4 B4 -2.3 87.0 C4 -1.0 51.2 D4 -2.0 41.0
5 104 A4 -0.0 5.6 B4 -2.3 86.9 C4 -0.8 48.6 D4 -1.2 29.1
1 104 A4 -0.0 6.4 B4 -0.6 74.8 C4 -0.3 38.1 D4 -1.2 18.9
5 102 A5 -0.0 87.6 B5 -0.7 86.6 C5 -2.6 79.7 D5 -5.6 75.2
1 102 A5 -0.0 12.2 B5 -0.7 86.6 C5 -1.5 71.5 D5 -4.9 75.1
5 103 A5 -0.0 12.5 B5 -0.7 86.6 C5 -0.4 68.1 D5 -4.9 75.2
1 103 A5 -0.0 12.4 B5 -0.7 86.6 C5 -0.2 53.2 D5 -4.7 67.6
5 104 A5 -0.0 12.3 B5 -0.6 86.3 C5 -0.0 46.8 D5 -3.8 60.0
1 104 A5 -0.0 12.5 B5 -0.1 19.0 C5 -0.0 33.2 D5 -3.3 49.8
5 102 A6 -0.9 87.2 B6 -0.8 86.6 C6 -3.3 79.9 D6 -7.9 76.0
1 102 A6 -0.9 87.2 B6 -0.8 86.6 C6 -2.0 70.5 D6 -5.9 74.8
5 103 A6 -0.9 87.2 B6 -0.8 86.6 C6 -1.3 65.4 D6 -5.0 74.0
1 103 A6 -0.8 87.1 B6 -0.7 86.3 C6 -0.2 49.6 D6 -2.5 71.1
5 104 A6 -0.5 86.8 B6 -0.1 72.1 C6 -0.2 39.9 D6 -2.5 71.2
1 104 A6 -0.0 66.1 B6 -0.0 7.6 C6 -0.0 36.6 D6 -2.5 67.3
5 102 A7 -0.0 87.5 B7 -3.1 86.2 C7 -3.8 74.4 D7 -6.5 75.5
1 102 A7 -0.0 11.7 B7 -3.1 86.2 C7 -2.4 65.7 D7 -5.3 72.1
5 103 A7 -0.0 11.7 B7 -3.1 86.2 C7 -1.9 60.7 D7 -4.0 68.0
1 103 A7 -0.0 11.3 B7 -3.1 86.2 C7 -0.8 43.0 D7 -2.8 61.2
5 104 A7 -0.0 11.5 B7 -3.0 86.0 C7 -0.0 36.4 D7 -2.2 60.6
1 104 A7 -0.0 11.4 B7 -0.8 75.8 C7 -0.0 14.0 D7 -2.2 57.4
5 102 A8 -0.0 88.1 B8 -1.5 86.7 C8 -3.6 73.9 D8 -11.5 76.2
1 102 A8 -0.0 88.1 B8 -1.5 86.7 C8 -3.3 74.7 D8 -6.7 73.4
5 103 A8 -0.0 88.1 B8 -1.5 86.7 C8 -3.3 74.4 D8 -6.7 73.4
1 103 A8 -0.0 16.4 B8 -1.2 86.4 C8 -3.3 73.7 D8 -4.4 68.2
5 104 A8 -0.0 16.6 B8 -0.8 74.5 C8 -3.2 65.6 D8 -3.4 67.9
1 104 A8 -0.0 16.5 B8 -0.0 7.8 C8 -2.2 60.5 D8 -2.4 64.9
5 102 A9 -0.0 88.0 B9 -1.9 85.9 C9 -4.1 75.3 D9 -2.1 75.2
1 102 A9 -0.0 88.0 B9 -1.9 85.9 C9 -2.7 74.8 D9 -2.1 75.2
5 103 A9 -0.0 88.0 B9 -1.9 85.9 C9 -1.1 74.4 D9 -2.1 75.2
1 103 A9 -0.0 16.0 B9 -1.9 85.9 C9 -1.1 66.6 D9 -2.1 75.2
5 104 A9 -0.0 16.0 B9 -1.7 84.9 C9 -0.2 56.5 D9 -2.1 67.7
1 104 A9 -0.0 15.9 B9 -0.5 45.2 C9 -0.2 55.7 D9 -1.9 60.4
5 102 A10 -0.0 83.3 B10 -0.3 87.7 C10 -0.4 76.3 D10 -7.1 73.7
1 102 A10 -0.0 75.4 B10 -0.3 87.6 C10 -0.4 76.2 D10 -6.9 73.8
5 103 A10 -0.0 0.5 B10 -0.3 87.7 C10 -0.3 67.9 D10 -6.4 73.1
1 103 A10 -0.0 -5.4 B10 -0.3 87.6 C10 -0.3 48.0 D10 -4.5 62.0
5 104 A10 -0.0 -4.8 B10 -0.0 87.4 C10 -0.3 48.0 D10 -4.3 57.3
1 104 A10 -0.0 -4.7 B10 -0.0 35.7 C10 -0.2 27.0 D10 -3.1 38.6
Number of stops
20
15
# of stops
10
5
0
0.05 0.01 0.005 0.001 5e04 1e04
Thresholds
Objective function gaps Time gaps

0
80
2
60
4
%gap
%gap
40
6
20
8
10
0
12
0.05 0.01 0.005 0.001 5e04 1e04 0.05 0.01 0.005 0.001 5e04 1e04
Thresholds Thresholds
Fig. 6. Experimental evaluation of the probabilistic stopping rule. In each boxplot,

the boxes represent the rst and the second quartile; solid line represent median while
dotted vertical line is the full variation range. Plots vary for each threshold . The dots
connected by a line represent the mean values.
The experimental evaluation of the probabilistic stop is summarized in the

three distinct boxplots of Fig. 6. Each boxplot reports a sensible information
related to the impact of the probabilistic stop, namely: the number of times
the probabilistic criterion has been satised, the gaps in the objective function
values, and the gaps in the computation times obtained comparing the solutions
obtained with and without the use of the probabilistic stopping rule. Such infor-
mation are collected, for each instance, as averages of the data obtained over 20
trials in the experimental setup described above. The rst boxplot depicts the
number of total stops recorded for dierent values of threshold . Larger values
of , indeed, yield a less coercive stopping rule, thus recording an higher number
of stops. Anyhow, even for the smallest, most conservative , the average number
of stops recorded is close to 50% of the tests performed. In the second boxplot,
the objective function gap is reported. Such gap quanties the qualitative wors-
ening in quality of the solutions obtained with the probabilistic stopping rule.
76 G. Felici et al.
Fig. 7. Comparison of objective function values and computation times obtained with
and without probabilistic stopping rule for dierent threshold values.
The gaps yielded show how even with the highest , the dierence in solution
quality is extremely small, with a single minimum of 11.5% for the instance D8,
and a very promising average gap, slightly below 2%. As expected, decreasing
the values the solutions obtained with and without the probabilistic stopping
rule will align with each other, and the negative gaps will accordingly grow up to
approximately 1%. The third boxplot shows the gaps obtained in the compu-
tation times. The analysis of such gaps is the key to realistically appraise the
actual benet provided by the use of the probabilistic stopping rule. Observing
the results reported, it is possible to note how even in the case of the smallest
threshold, i.e., using the most strict probabilistic stopping criterion, the stops
recorded are such that an average time discount close to the 40% is encountered.
A more direct display of this time gaps can be obtained straightly considering
the total time discount in seconds: with the smallest we have experienced a
time discount of 4847.6 s over the 11595.9 total seconds needed for the execution
without the probabilistic stop. Analyzing in the same fashion the values obtained
under the largest threshold, we observed an excellent average discount just over
80%, which quantied in seconds amounts to an astonishing total discount of
8919.64 s over the 11595.9 total seconds registered for the execution without the
probabilistic stop.
6 Conclusions
In this paper, we have investigated a strategy for a GRASP heuristic that solves
large sized MinCostSAT. The method adopts a straight-forward implementation
of the main ingredients of the heuristic, but proposes a new probabilistic stopping
rule. Experimental results show that, for instances belonging to particular class
of MinCostSAT problems, the method performs very well and the new stopping
rule provides a very eective way to reduce the number of iterations of the
algorithm without observing any signicant decay in the value of the objective
function.
The work presented has to be considered preliminary, but it clearly indi-
cates several research directions that we intend to pursue: the renement of the
dynamic estimate of the probability distribution of the solutions found by the
algorithm, the comparative testing of instances of larger size, and the extension
to other classes of problems. Last, but not least, attention will be directed toward
the incorporation of the proposed heuristic into methods that are specically
designed to extract logic formulas from data and to the test of the performances
of the proposed algorithm in this setting.
Acknowledgements. This work has been realized thanks to the use of the S.Co.P.E.
computing infrastructure at the University of Napoli FEDERICO II.
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A New Local Search for the p-Center Problem
Based on the Critical Vertex Concept
Daniele Ferone1 , Paola Festa1(B) , Antonio Napoletano1 ,

and Mauricio G.C. Resende2
1
Department of Mathematics and Applications, University of Napoli Federico II,
Compl. MSA, Via Cintia, 80126 Naples, Italy
{daniele.ferone,paola.festa,antonio.napoletano2}@unina.it
2
Mathematical Optimization and Planning, Amazon.com, Seattle, USA
resendem@amazon.com
Abstract. For the p-center problem, we propose a new smart local

search based on the critical vertex concept and embed it in a GRASP
framework. Experimental results attest the robustness of the proposed
search procedure and conrm that for benchmark instances it converges
to optimal or near/optimal solutions faster than the best known state-
of-the-art local search.
1 Introduction
The p-center problem is one of the best-known discrete location problems rst
introduced in the literature in 1964 by Hakimi [13]. It consists of locating p facil-
ities and assigning clients to them in order to minimize the maximum distance
between a client and the facility to which the client is assigned (i.e., the clos-
est facility). Useless to say that this problem arises in many dierent real-world
contexts, whenever one designs a system for public facilities, such as schools or
emergency services.
Formally, we are given a complete undirected edge-weighted bipartite graph
G = (V U, E, c), where
V = {1, 2, . . . , n} is a set of n potential locations for facilities;

U = {1, 2, . . . , m} is a set of m clients or demand points;
E = {(i, j)| i V, j U } is a set of n m edges;
c : E R+ {0} is a function that assigns a nonnegative distance cij to each
edge (i, j) E.
The p-center problem is to nd a subset P V of size p such that its weight,

dened as
C(P ) = max min cij (1)

iU jP
is minimized. The minimum value is called the radius. Although it is not a

restrictive hypothesis, in this paper we consider the special case where V = U
https://doi.org/10.1007/978-3-319-69404-7_6
80 D. Ferone et al.
is the vertex set of a complete graph G = (V, E), each distance cij represents
the length of a shortest path between vertices i and j (cii = 0), and hence the
triangle inequality is satised.
In 1979, Kariv and Hakimi [16] proved that the problem is NP -hard, even in
the case where the input instance has a simple structure (e.g., a planar graph
of maximum vertex degree 3). In 1970, Minieka [20] designed the rst exact
method for the p-center problem viewed as a series of set covering problems. His
algorithm iteratively chooses a threshold r for the radius and checks whether all
clients can be covered within distance r using no more than p facilities. If so, the
threshold r is decreased; otherwise, it is increased. Inspired by Miniekas idea,
in 1995 Daskin [3] proposed a recursive bisection algorithm that systematically
reduces the gap between upper and lower bounds on the radius. More recently,
in 2010 Salhi and Al-Khedhairi [26] proposed a faster exact approach based on
Daskins algorithm that obtains tighter upper and lower bounds by incorporat-
ing information from a three-level heuristic that uses a variable neighborhood
strategy in the rst two levels and at the third level a perturbation mechanism
for diversication purposes.
Recently, several facility location problems similar to the p-center have been
used to model scenarios arising in nancial markets. The main steps to use such
techniques are the following: rst, to describe the considered nancial market
via a correlation matrix of stock prices; second, to model the matrix as a graph,
stocks and correlation coecients between them are represented by nodes and
edges, respectively. With this idea, Goldengorin et al. [11] used the p-median
problem to analyze stock markets. Another interesting area where these problems
arise is the manufacturing system with the aim of lowering production costs [12].
Due to the computational complexity of the p-center problem, several approx-
imation and heuristic algorithms have been proposed for solving it. By exploiting
the relationship between the p-center problem and the dominating set prob-
lem [15,18], nice approximation results were proved. With respect to inapprox-
imability results, Hochbaum and Shmoys [15] proposed a 2-approximation algo-
rithm for the problem with triangle inequality, showing that for any < 2 the
existence of a -approximation algorithm would imply that P = NP .
Although interesting in theory, approximation algorithms are often outper-
formed in practice by more straightforward heuristics with no particular perfor-
mance guarantees. Local search is the main ingredient for most of the heuristic
algorithms that have appeared in the literature. In conjunction with various tech-
niques for escaping local optima, these heuristics provide solutions which exceed
the theoretical upper bound of approximating the problem and derive from ideas
used to solve the p-median problem, a similar NP -hard problem [17]. Given a set
F of m potential facilities, a set U of n users (or customers), a distance function
d : U F R, and a constant p m, the p-median problem is to determine
a subset of p facilities to open so as to minimize the sum of the distances from
each user to its closest open facility. For the p-median problem, in 2004 Resende
and Werneck [25] proposed a multistart heuristic that hybridizes GRASP with
Path-Relinking as both, intensication and post-optimization phases. In 1997,
A New Local Search for the p-Center Problem 81
Hansen and Mladenovic [14] proposed three heuristics: Greedy, Alternate, and
Interchange (vertex substitution). To select the rst facility, Greedy solves a
1-center problem. The remaining p1 facilities are then iteratively added, one at
a time, and at each iteration the location which most reduces the maximum cost
is selected. In [5], Dyer and Frieze suggested a variant, where the rst center is
chosen at random. In the rst iteration of Alternate, facilities are located at p
vertices chosen in V , clients are assigned to the closest facility, and the 1-center
problem is solved for each facilitys set of clients. During the subsequent itera-
tions, the process is repeated with the new locations of the facilities until no more
changes in assignments occur. As for the Interchange procedure, a certain pat-
tern of p facilities is initially given. Then, facilities are moved iteratively, one by
one, to vacant sites with the objective of reducing the total (or maximum) cost.
This local search process stops when no movement of a single facility decreases
the value of the objective function. A multistart version of Interchange was also
proposed, where the process is repeated a given number of times and the best
solution is kept. The combination of Greedy and Interchange has been most
often used for solving the p-median problem. In 2003, Mladenovic et al. [21]
adapted it to the p-center problem and proposed a Tabu Search (TS) and a
Variable Neighborhood Search (VNS), i.e., a heuristic that uses the history of
the search in order to construct a new solution and a competitor that is not
history sensitive, respectively. The TS is designed by extending Interchange to
the chain-interchange move, while in the VNS, a perturbed solution is obtained
from the incumbent by a k-interchange operation and Interchange is used to
improve it. If a better solution than the incumbent is found, the search is recen-
tered around it. In 2011, Davidovic et al. [4] proposed a Bee Colony algorithm, a
random search population-based technique, where an articial system composed
of a number of precisely dened agents, also called individuals or articial bees.
To the best of our knowledge, most of the research eort devoted towards
the development of metaheuristics for this problem has been put into the design
of ecient local search procedures. The purpose of this article is propose a new
local search and to highlight how its performances are better than best-known
local search proposed in literature (Mladenovic et al.s [21] local search based on
VNS strategy), both in terms of solutions quality and convergence speed.
The remainder of the paper is organized as follows. In Sect. 2, a GRASP con-
struction procedure is described. In Sect. 3, we introduce the new concept of
critical vertex with relative denitions and describe a new local search algorithm.
Computational results presented in Sect. 4 empirically demonstrate that our local
search is capable of obtaining better results than the best known local search,
and they are validated by a statistical signicance test. Concluding remarks are
made in Sect. 5.
2 GRASP Construction Phase

GRASP is a randomized multistart iterative method proposed in Feo and
Resende [6,7] and having two phases: a greedy randomized construction phase
82 D. Ferone et al.
and a local search phase. For a comprehensive study of GRASP strategies and
their variants, the reader is referred to the survey papers by Festa and Resende
[9,10], as well as to their annotated bibliography [8].
Starting from a partial solution made of 1 randElem p facilities ran-
domly selected from V , our GRASP construction procedure iteratively selects
the remaining prandElem facilities in a greedy randomized fashion. The greedy
function takes into account the contribution to the objective function achieved
by selecting a particular candidate element. In more detail, given a partial solu-
tion P , |P | < p, for each i V \ P , we compute w(i) = C(P {i}). The pure
greedy choice would consist in selecting the vertex with the smallest greedy func-
tion value. This procedure instead computes the smallest and the largest greedy
function values:
zmin = min w(i); zmax = max w(i).

iV \P iV \P
Then, denoting by = zmin + (zmax zmin ) the cut-o value, where is a

parameter such that [0, 1], a restricted candidate list (RCL) is made up of
all vertices whose greedy value is less than or equal to . The new facility to be
added to P is nally randomly selected from the RCL.
The pseudo-code is shown in Fig. 1, where [0, 1].
Function greedy-randomized-build(G = V, E, C , p, , )

1 P ;
2 randElem := p ;
3 for k = 1, . . . , randElem do // random component
4 f SelectRandom(V \ P );
5 P P {f } ;
6 while |P | < p do
7 zmin + ;
8 zmax ;
9 for i V \ P do
10 if zmin > C(P {i}) then
11 zmin C(P {i}) ;
12 if zmax < C(P {i}) then
13 zmax C(P {i}) ;
14 zmin + (zmax zmin ) ;
15 RCL {i V \ P | C(P {i}) } ;
16 f SelectRandom(RCL) ;
17 P P {f };
18 return P ;
Fig. 1. Pseudo-code of the greedy randomized construction.

3 Plateau Surfer: A New Local Search Based

on the Critical Vertex Concept
Given a feasible solution P , the Interchange local search proposed by Hansen

and Mladenovic [14] consists in swapping a facility f P with a facility f /P
which results in a decrease of the current cost function. Especially in the case of
instances with many vertices, we have noticed that usually a single swap does not
strictly improve the current solution, because there are several facilities whose
distance is equal to the radius of the solution. In other words, the objective
function is characterized by large plateaus and the Interchange local search
cannot escape from such regions. To face this type of diculties, inspired by
Variable Formulation Search [22,23], we have decided to use a rened way for
comparing between valid solutions by introducing the concept of critical vertex.
Given a solution P V , let P : V R+ {0} be a function that assigns
to each vertex i V the distance between i and its closest facility according
to solution P . Clearly, the cost of a solution P can be equivalently written as
C(P ) = max{P (i) : i V }. We also give the following denition:
Definition 1 (Critical vertex). Let P V be a solution whose cost is C(P ).

For each vertex i V , i is said to be a critical vertex for P , if and only if
P (i) = C(P ).
In the following, we will denote with maxP = |{i V : P (i) = C(P )}| the
number of vertices whose distance from their closest facility results in the objec-
tive function value corresponding to solution P . We dene also the comparison
operator <cv , and we will say that P <cv P if and only if maxP < maxP .
N (y2 ) N (y2 )
P P
y2 y2
N (y1 )
N (y1 )
xl1 xl1
xj1 xj1 N (y3 )
xi1
N (y3 ) xi1
C(P ) y1 y3 C(P ) y1 y3
xk1 xk1
Fig. 2. An example of how the local search works. In this case, the algorithm switches
from solution P to solution P . In P , a new facility y3 is selected in place of y3 in
P , y3 attracts one of the critical vertices from the neighborhood of the facility y1 .
Although the cost of the two solutions is the same, the algorithm selects the new
solution P because maxP < maxP .
84 D. Ferone et al.
The main idea of our plateau surfer local search is to use the concept of
critical vertex to escape from plateaus, moving to solutions that have either
a better cost than the current solution or equal cost but less critical vertices.
Figure 2 shows a simple application of the algorithm, while in Figs. 3 and 4, for
four benchmark instances, both Mladenovics local search and our local search
are applied once taken as input the same starting feasible solution. It is evident
that both the procedures make the same rst moves. However, as soon as a
plateau is met, Mladenovics local search ends, while our local search is able to
escape from the plateau moving to other solutions with the same cost value,
Fig. 3. Plateau escaping. The behavior of our plateau surfer local search (in red) com-
pared with the Mladenovics one (in blue). Both algorithms work on the same instances
taking as input the same starting solution. Filled red dots and empty blue circles indi-
cate the solutions found by the two algorithms. Mladenovic local search stops as soon
as the rst plateau is met. (Color gure online)
Fig. 4. Plateau escaping. The behavior of our plateau surfer local search (in red) com-
pared with the Mladenovics one (in blue) on other two dierent instances. (Color gure
online)
and restarting the procedure from a new solution that can lead to a strict cost
function improvement.
Let us analyze in more detail the behavior of our local search, whose pseudo-
code is reported in Fig. 5. The main part of the algorithm consists in the portion of
the pseudo-code that goes from line 7 to line 14. Starting from an initial solution
P , the algorithm tries to improve the solution replacing a vertex j / P with a
facility i P . Clearly, this swap is stored as an improving move if the new solution
P = P \{i}{j} is strictly better than P according to the cost function C. If C(P )
is better than the current cost C(P ), then P is compared also with the incumbent
86 D. Ferone et al.
Function plateau-surfer-local-search(G = V, A, C , P, p)

1 repeat
2 modified := false;
3 forall i P do
4 best flip := best cv flip := NIL;
5 bestNewSolValue := C(P );
6 best cv := max (P );
7 forall j V \ P do
8 P := P \ {i} {j};
9 if C(P ) < bestNewSolValue then
10 bestNewSolValue := C(P );
11 best flip := j;
12 else if best flip = NIL and max (P ) < best cv then
13 best cv := max (P );
14 best cv flip := j;
15 if best flip
= NIL then
16 P := P \ {i} {best flip};
17 modified := true;
18 else if best cv flip
= NIL then
19 P := P \ {i} {best cv flip};
20 modified := true;
21 until modified = false;
22 return P ;
Fig. 5. Pseudocode of the plateau surfer local search algorithm based on the critical
vertex concept.
solution and if it is the best solution found so far, the incumbent is update and
the swap that led to this improvement stored (lines 911).
Otherwise, the algorithm checks if it is possible to reduce the number of
critical vertices. If the new solution P is such that P <cv P , then the algorithm
checks if P is the best solution found so far (line 12), the value that counts the
number of critical vertices in a solution is update (line 13), and the current swap
stored as an improving move (line 14).
To study the computational complexity of our local search, let be n = |V |
and p = |P |, the number of vertices in the graph and the number of facilities in
a solution, respectively. The loops at lines 3 and 7 are executed p and n times,
respectively. The update of the solution takes O(n). In conclusion, the total
complexity is O(p n2 ).
4 Experimental Results
In this section, we describe computational experience with the local search pro-
posed in this paper. We have compared it with the local search proposed by
Mladenovic et al. [21], by embedding both in a GRASP framework.
The algorithms were implemented in C++, compiled with gcc 5.2.1 under
Ubuntu with -std=c++14 ag. The stopping criterion is maxT ime = 0.1n+0.5
p. All the tests were run on a cluster of nodes, connected by 10 Gigabit Inniband
technology, each of them with two processors Intel Xeon E5-4610v2@2.30 GHz.
Table 1 summarizes the results on a set of ORLIB instances, originally intro-
duced in [1]. It consists of 40 graphs with number of vertices ranging from 100 to
900, each with a suggested value of p ranging from 5 to 200. Each vertex is both
a user and a potential facility, and distances are given by shortest path lengths.
Tables 2 and 3 report the results on the TSP set of instances. They are just sets
of points on the plane. Originally proposed for the traveling salesman problem,
they are available from the TSPLIB [24]. Each vertex can be either a user or
an open facility. We used the Mersenne Twister random number generator by
Matsumoto and Nishimura [19]. Each algorithm was run with 10 dierent seeds,
and minimum (min), average (E) and variance ( 2 ) values are listed in each
table. The second to last column lists the %-Gap between average solutions. To
deeper investigate the statistical signicance of the results obtained by the two
local searches, we performed the Wilcoxon test [2,27].
Generally speaking, the Wilcoxon test is a ranking method that well applies
in the case of a number of paired comparisons leading to a series of dierences,
some of which may be positive and some negative. Its basic idea is to substitute
scores 1, 2, 3, . . . , n with the actual numerical data, in order to obtain a rapid
approximate idea of the signicance of the dierences in experiments of this
kind.
More formally, let A1 and A2 be two algorithms, I1 , . . . , Il be l instances of
the problem to solve, and let Ai (Ij ) be the value of the solution obtained by
algorithm Ai (i = 1, 2) on instance Ij (j = 1, . . . , l). For each j = 1, . . . , l, the
Wilcoxon test computes the dierences j = |A1 (Ij ) A2 (Ij )| and sorts them
in non decreasing order. Accordingly, starting with a smallest rank equal to 1, to
each dierence j , it assigns a non decreasing rank Rj . Ties receive a rank equal
to the average of the sorted positions they span. Then, the following quantities
are computed

W+ = Rj ,
j=1,...,l : j >0

W = Rj .
j=1,...,l : j <0
Under the null hypothesis that A1 (Ij ) and A2 (Ij ) have the same median
value, it should result that W + = W . If the p-value associated to the experi-
ment is less than an a priori xed signicance level , then the null hypothesis
is rejected and the dierence between W + and W is considered signicant.
The last column of each table lists the p-values where the %-Gap is signicant,
all the values are less than = 0.01. This outcome of the Wilcoxon test further
conrms that our local search is better performing than the local search proposed
by Mladenovic et al.
88 D. Ferone et al.
Table 1. Results on ORLIB instances.
Instance GRASP + mladenovic GRASP + plateau-surfer %-Gap p-value

min E 2 min E 2
pmed01 127 127 0 127 127 0 0.00
pmed02 98 98 0 98 98 0 0.00
pmed03 93 93.14 0.12 93 93.54 0.25 0.43
pmed04 74 76.21 1.33 74 74.02 0.04 2.87 1.20E16
pmed05 48 48.46 0.43 48 48 0 0.95
pmed06 84 84 0 84 84 0 0.00
pmed07 64 64.15 0.27 64 64 0 0.23
pmed08 57 59.39 1.36 55 55.54 0.73 6.48 3.37E18
pmed09 42 46.87 2.83 37 37.01 0.01 21.04 2.80E18
pmed10 29 31.21 0.81 20 20.01 0.01 35.89 9.38E19
pmed11 59 59 0 59 59 0 0.00
pmed12 51 51.89 0.1 51 51.41 0.24 0.93
pmed13 42 44.47 0.73 36 36.94 0.06 16.93 1.04E18
pmed14 35 38.59 3.24 26 26.85 0.13 30.42 2.11E18
pmed15 28 30.23 0.7 18 18 0 40.46 1.09E18
pmed16 47 47 0 47 47 0 0.00
pmed17 39 40.71 0.23 39 39 0 4.20 8.69E20
pmed18 36 37.95 0.29 29 29.41 0.24 22.50 6.37E19
pmed19 27 29.32 0.42 19 19.13 0.11 34.75 6.25E19
pmed20 25 27.05 0.99 14 14 0 48.24 1.46E18
pmed21 40 40 0 40 40 0 0.00
pmed22 39 40.06 0.24 38 38.94 0.06 2.80 1.30E18
pmed23 30 32.02 0.44 23 23.21 0.17 27.51 7.16E19
pmed24 24 25.38 0.34 16 16 0 36.96 4.37E19
pmed25 22 22.62 0.24 11 11.89 0.1 47.44 2.77E19
pmed26 38 38 0 38 38 0 0.00
pmed27 33 33.96 0.06 32 32 0 5.77 2.15E22
pmed28 26 26.78 0.17 19 19 0 29.05 2.20E20
pmed29 23 23.43 0.31 13 13.68 0.22 41.61 8.00E19
pmed30 20 21.18 0.47 10 10 0 52.79 6.50E19
pmed31 30 30 0 30 30 0 0.00
pmed32 30 30.37 0.23 29 29.62 0.24 2.47
pmed33 23 23.76 0.2 16 16.28 0.2 31.48 4.31E19
pmed34 21 22.42 0.66 11 11.56 0.25 48.44 1.59E18
pmed35 30 30.01 0.01 30 30 0 0.03
pmed36 28 29.37 0.31 27 27.65 0.23 5.86 4.52E18
pmed37 23 24.07 0.37 16 16 0 33.53 2.74E19
pmed38 29 29 0 29 29 0 0.00
pmed39 24 25.08 0.11 23 23.98 0.02 4.39 4.68E21
pmed40 20 21.81 0.43 14 14 0 35.81 5.14E19
Average 16.78
Table 2. Results on TSPLIB instances (1)
Instace p GRASP + mladenovic GRASP + plateau-surfer %-Gap p-value

min E 2 min E 2
pcb3038 50 534.48 608.49 1068.09 355.68 374.66 51.05 38.43 3.90E18
100 399.49 481.75 1285.58 259.67 270.2 17.56 43.91 3.90E18
150 331.62 428.69 1741.11 206.71 215.78 23.73 49.67 3.90E18
200 301.01 386.56 3161.87 177.79 190.88 10.4 50.62 3.90E18
250 292.48 359.59 3323.62 155.03 163.75 19.24 54.46 3.90E18
300 261.28 349.42 2902.71 143.39 151.89 10.04 56.53 3.90E18
350 258.82 336.08 3755.72 123.85 136.22 22.45 59.47 3.90E18
400 249.78 337.14 4033.46 119.07 122.31 1.2 63.72 3.90E18
450 214.97 321.36 3373.23 115 117 0.6 63.59 3.90E18
500 209.35 299.4 3378.98 102 110.38 5.78 63.13 3.90E18
pr1002 10 3056.55 3313.49 10132.77 2616.3 2727.45 2260.95 17.69 3.90E18
20 2404.16 2668.29 8244.65 1806.93 1886.89 1516.07 29.28 3.90E18
30 2124.26 2358.07 4432.11 1456.02 1505.55 910.93 36.15 3.89E18
40 1960.23 2172.63 7831.77 1253.99 1302.76 751.62 40.04 3.90E18
50 1755.7 1992.08 5842.66 1097.72 1156.77 815.35 41.93 3.90E18
60 1697.79 1865.5 5872.47 1001.25 1042.82 257.42 44.1 3.89E18
70 1569.24 1736.41 4078.39 900 954.04 307.65 45.06 3.89E18
80 1486.61 1633.87 3278.4 851.47 889.5 407.29 45.56 3.88E18
90 1350.93 1543.17 3922.25 764.85 809.78 382.29 47.52 3.89E18
100 1312.44 1472.47 2616 743.3 767.62 77.4 47.87 3.89E18
pr439 10 2575.12 2931.83 38470.59 1971.83 1972.28 19.61 32.73 3.79E18
20 1940.52 2577.03 23638.88 1185.59 1194.12 124.58 53.66 3.71E18
30 1792.34 2510.91 23692.47 886 919.1 442.37 63.4 3.89E18
40 1525.2 2413.33 53876.4 704.45 728.19 39.31 69.83 3.88E18
50 1358.54 2252.46 89633.71 575 595.4 64.21 73.57 3.82E18
60 1386.09 2170.85 110065.93 515.39 537.66 75.43 75.23 3.89E18
70 1370.45 1898.53 116167.77 480.23 499.65 4.93 73.68 3.73E18
80 1140.18 1815.1 118394.68 424.26 440.27 166.06 75.74 3.89E18
90 1191.9 1699.64 91388.99 400 406.17 31.71 76.1 3.88E18
100 1190.85 1679.73 94076.45 375 384.27 98.91 77.12 3.89E18
rat575 10 81.32 92.98 9.27 73 74.71 0.79 19.65 3.90E18
20 68.07 73.86 3.7 50.54 53.04 0.63 28.19 3.90E18
30 59.81 64.61 3.67 41.79 43.53 0.47 32.63 3.90E18
40 54.13 58.37 3.43 36.12 37.43 0.29 35.87 3.90E18
50 47.68 53.78 3.56 32.45 33.36 0.17 37.97 3.90E18
60 45.62 50.03 3.21 29.15 30.17 0.19 39.7 3.90E18
70 43.68 46.96 2.97 27 27.78 0.13 40.84 3.90E18
80 39.81 44.2 2.75 25.02 25.99 0.11 41.2 3.90E18
90 38.48 41.98 2.06 23.85 24.4 0.07 41.88 3.90E18
100 37.01 39.93 1.4 22.2 23.01 0.08 42.37 3.89E18
rat783 10 102.22 110.93 13.17 83.49 87.82 1.65 20.83 3.90E18
20 80.53 88.56 7.68 59.68 62.8 1.41 29.09 3.90E18
30 69.58 76.92 7.87 49.25 51.48 0.74 33.07 3.90E18
40 62.97 69.63 4.62 42.05 44.27 0.53 36.42 3.90E18
50 59.41 65.26 5.59 38.29 39.6 0.42 39.32 3.90E18
60 54.82 60.35 4.77 34.48 35.92 0.24 40.48 3.90E18
70 49.4 56.56 7.48 32.06 33.11 0.24 41.46 3.90E18
80 48.51 53.76 4.03 29.55 30.94 0.21 42.45 3.90E18
90 46.07 51.82 3.53 28.18 28.85 0.11 44.33 3.90E18
100 43.97 49.5 4.68 26.31 27.49 0.14 44.46 3.90E18
Average 46.84
90 D. Ferone et al.
Table 3. Results on TSPLIB instances (2)
Instance p GRASP + mladenovic GRASP + plateau-surfer %-Gap p-value

min E 2 min E 2
rl1323 10 3810.84 4185.89 24655.46 3110.57 3241.79 3290.56 22.55 3.90E18
20 2996.4 3348.31 23183.21 2090.87 2236.28 2798.56 33.21 3.90E18
30 2689.44 2979.79 14205.75 1730.78 1808.94 1544.85 39.29 3.90E18
40 2337.92 2712.93 14193.05 1479.24 1576.25 1710.4 41.9 3.90E18
50 2195.91 2462.95 9835.09 1300 1363.88 950.66 44.62 3.90E18
60 2021.87 2278.94 16400.27 1181.3 1244.03 657.55 45.41 3.90E18
70 1900.77 2128.45 11883.58 1076.2 1127.98 475.13 47 3.90E18
80 1866.8 2033.24 4501.73 988.87 1048.87 438.82 48.41 3.89E18
90 1634.37 1966.13 4643.42 935.02 978.6 289.18 50.23 3.89E18
100 1631.5 1909.56 8483.55 886.85 914 238.2 52.14 3.89E18
u1060 10 3110.65 3373.87 7541.61 2301.7 2440 599.42 27.68 3.86E18
20 2652.6 2818.37 5787.51 1650.34 1749.15 2814.03 37.94 3.90E18
30 2501.72 2684.87 3811.23 1302.94 1373.21 912.92 48.85 3.90E18
40 2442.07 2616.15 5267.85 1118.59 1176.14 593.6 55.04 3.90E18
50 2378.36 2591.96 7266.77 950.66 1021.55 418.91 60.59 3.90E18
60 2301.83 2602.13 13579.82 860.49 919.97 374.54 64.65 3.90E18
70 2378.36 2606.64 10944.09 790.13 828.16 441.03 68.23 3.90E18
80 2351.82 2622.32 12980.39 720.94 753.64 306.94 71.26 3.90E18
90 2248.61 2562.01 10260.36 667.55 708.04 107.79 72.36 3.90E18
100 2060.29 2494.08 11025.91 632.11 653.15 110.65 73.81 3.90E18
110 2049.18 2444.22 10385.95 570.49 613.02 148.7 74.92 3.90E18
120 2122.97 2406.19 9191.4 570 579.93 96.23 75.9 3.90E18
130 1839.55 2390.82 12029.95 538.82 561.62 78.78 76.51 3.90E18
140 1924.48 2316.25 12982.87 500.39 527.66 172.51 77.22 3.90E18
150 1942.27 2300.45 13245.06 499.65 503.26 20.49 78.12 3.90E18
u1817 10 592.97 646.89 325 466.96 485.44 104.33 24.96 3.90E18
20 462.3 564.44 560.9 330.2 348.15 53.96 38.32 3.90E18
30 418.91 530.34 1018.29 265.19 283.4 58.43 46.56 3.90E18
40 407.19 526.44 956.01 232.25 245.78 43.42 53.31 3.90E18
50 330.21 507.52 2889.76 204.79 217.05 26.96 57.23 3.90E18
60 352.88 497.35 3539.09 184.91 197.26 21.79 60.34 3.90E18
70 321.27 477.43 4139.93 170.39 181.53 13.67 61.98 3.90E18
80 289.61 445.35 4866.81 154.5 166.46 22.68 62.62 3.90E18
90 283.99 422.34 3828.2 148.11 153.5 13.72 63.65 3.90E18
100 283.99 416.69 2660.21 136.79 146.67 7.4 64.8 3.90E18
Average 54.9
5 Concluding Remarks
In this paper, we presented a new local search heuristic for the p-center problem,
whose potential applications appear in telecommunications, in transportation
logistics, and whenever one must to design a system to organize some sort of
public facilities, such as, for example, schools or emergency services.
The computational experiments show that the proposed local search is capa-
ble to reduce the number of local optimum solutions using the concept of critical
vertex, and it improves the results of the best local search for the problem.
Future lines of work will be focused on a deeper investigation of the robustness
of our proposal by applying it on further instances coming from nancial markets
and manufacturing systems.
Acknowledgements. This work has been realized thanks to the use of the S.Co.P.E.
computing infrastructure at the University of Napoli FEDERICO II.
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An Iterated Local Search Framework with
Adaptive Operator Selection for Nurse Rostering
Angeliki Gretsista(B) and Edmund K. Burke
School of Electronic Engineering and Computer Science,

Queen Mary University of London, London, UK
{a.gretsista,vp-se}@qmul.ac.uk
Abstract. Considerable attention has been paid to selective hyper-

heuristic frameworks for addressing computationally hard scheduling
problems. By using selective hyper-heuristics, we can derive benets from
the strength of low level heuristics and their components at dierent
stages of the heuristic search. In this paper, a simple, general and eec-
tive selective hyper heuristic is presented. We introduce an iterated local
search based hyper-heuristic framework that incorporates the adaptive
operator selection scheme to learn through the search process. The con-
sidered iterative approach employs an action selection model to decide
the perturbation strategy to apply in each step and a credit assignment
module to score its performance. The designed framework allows us to
employ any action selection model and credit assignment mechanism
used in the literature. Empirical results and an analysis of six dier-
ent action selection models against state-of-the-art approaches, across
39 problem instances, highlight the signicant potential of the proposed
selection hyper-heuristics. Further analysis on the adaptive behavior of
the model suggests that two of the six models are able to learn the best
performing perturbation strategy, resulting in signicant performance
gains.
Keywords: Nurse rostering Personnel scheduling Hyper-heuristics

Action selection models
1 Introduction
One of the main motivations for the development of hyper-heuristics was to
develop search algorithms that can operate with a certain degree of gen-
erality [1,2]. Hyper-heuristics can be considered to be high level general
approaches that are able to select or generate low-level heuristics, whilst restrict-
ing the need to use domain knowledge [3]. In particular, selection hyper-heuristics
choose heuristics from a predened set of low level heuristics within a framework,
where the aim is to determine a sequence of perturbations that provide ecient
solutions for a given problem. On the other hand, the idea behind generative
hyper-heuristics is to develop new heuristics based on the basic components of
the input low level heuristics [3].
https://doi.org/10.1007/978-3-319-69404-7_7
94 A. Gretsista and E.K. Burke
Nurse rostering is a well-known and well-studied personnel scheduling prob-

lem. The main objective is to assign the appropriate level of qualied nurses
to shifts to cover the demand across dierent medical wards in a hospital, with
respect to a diverse set of hard and soft constraints [4]. Representative constraints
capture work regulations, employee preferences and fairness of allocation regard-
ing weekend/night shifts or leave and days o [5]. To address nurse rostering
problems, researchers have proposed a wide variety of methodologies including
hyper-heuristics. In the recent work of Asta et al. [6], a tensor based algorithm
is embodied to an online learning selection hyper-heuristic framework. In [7] Lu
and Hao present an algorithm that takes two neighborhood moves and adaptively
switches between search strategies by considering their search history. Other
recent approaches constitute evolutionary perturbative hyper-heuristics [8], har-
mony search-based hyper-heuristics [9] and hybrid approaches [1012].
Several publications have appeared in recent years proposing selective hyper-
heuristics to address problems across dierent domains. Representative works
can be the sequence-based selection hyper heuristic, which is motivated by
the hidden Markov model [13], the PHunter [14], the Fair-Share Iterated Local
Search [15], as well as the GIHH [16], the winner of the CHeSC 2011 [17] com-
petition. The latter, is a characteristic example of a quite eective, yet highly
complex, state-of-the-art hyper-heuristic. GIHH incorporates a dynamic heuris-
tic selection mechanism and a move acceptance strategy, which evolves based on
the properties of the current search landscape. We can observe that many of the
best performing selective hyper-heuristics are developed based on the Iterated
Local Search (ILS) algorithm. The ILS algorithm is driven by the alternation
between intensication, intensive search of the neighborhood for locating locally
optimal solutions, and diversication, in order to jump to new unexplored
neighborhoods that can lead to better local optima. Determining the appropri-
ate amount of diversication re-actively is essential for eective search [18].
In this study, we incorporate the adaptive operator selection paradigm in
the main structure of an ILS algorithm to re-actively identify the most eec-
tive perturbation strategy. The proposed approach is a generic hyper-heuristic
framework, namely the Iterated Local Search based hyper-heuristic framework
(HHILS). HHILS incorporates an action selection mechanism to learn and adap-
tively select the most ecient low-level perturbation heuristic and a credit assign-
ment module to empirically estimate the quality of the selected perturbation
according to its ability to improve the incumbent solution. We utilize six state-
of-the-art action selection models that result in six HHILS variants and compare
them to eight state-of-the-art hyper-heuristics, on a set of 39 challenging nurse
problem instances. The experimental results clearly justify the potential of the
proposed framework.
The remainder of the paper is organized as follows: The nurse rostering prob-
lem is briey described in Sect. 2. The proposed hyper-heuristic framework is
presented in Sect. 3 where the main algorithm with the credit assignment and
the action selection procedures are briey explained in Sects. 3.1 and 3.2 respec-
tively. Section 4 presents the experimental setup in Sect. 4.1 and the results in
An Iterated Local Search Framework with Adaptive Operator Selection 95
Sect. 4.2. Finally, Sect. 5 concludes with a summary of the experimental ndings
of this work and provides some pointers for future work.
2 The Nurse Rostering Problem

In this paper, we use the model proposed in [10] to address the nurse rostering
problem. To capture a large variety of dierent constraints the model adopts
only one hard constraint, i.e., each employee can be assigned only one shift
per day. It incorporates the remaining constraints as soft constraints in the
objective function. Thus, the objective function is a weighted sum of the soft
constraints, in which the associated weights take high or low values depending
on their importance. Note that, associating a very high weight to a constraint
is analogous to considering it as hard, whilst associating a very low value is
analogous to discarding the specic constraint. This formulation also facilitates
the easy integration of additional constraints based on the needs of dierent
hospitals.
The soft constrains can be categorized in coverage constraints, where two
or more nurses are associated, and in employee working constraints, that refer
to each employee separately. Mainly, the coverage constraints ensure that the
required number of employees is working over each shift. Coverage constraints
can be also exploited to select skilled or qualied sta for specic shifts. Likewise,
the employees working constraints refer to the preference of each employee. The
aim at this point is to satisfy the individual nurse by being exible to decide
her own schedule. For instance, we aim to be able to identify specic workload,
to incorporate preferred days o, to select the most convenient shift pattern, to
dene vacations and so on.
3 The Proposed Approach

The proposed framework incorporates the adaptive operator selection paradigm
[19] within the main structure of an ILS algorithm. HHILS replaces the appli-
cation of a single perturbation heuristic of the ILS algorithm with an action
selection mechanism that adaptively selects among various low-level perturba-
tion heuristics. The applied perturbation heuristic is followed by the local search
and acceptance procedures of ILS and then is scored based on its ability to
improve the incumbent solution. As such, the proposed framework adaptively
learns which is the most eective perturbation heuristic to apply during search.
A description of the framework and its main characteristics can be briey out-
lined as follows.
Algorithm 1 demonstrates the main algorithmic steps of the developed frame-
work. As a rst step, HHILS initializes all the required modules and data struc-
tures. This step also comprises the input of the chosen action selection and the
credit assignment module (line 1). Next, HHILS initializes a solution for the
given problem instance (line 2). The initial solution can be randomly gener-
ated or constructed based on a constructive heuristic, depending on the problem
Algorithm 1. Pseudo code of the proposed HHILS framework.

1: Initialise the action selection, and the credit assignment module as well as create
all data structures required by HHILS.
2: scur GenerateInitialSolution() /* Generate Initial Solution: Initialise or con-
struct a solution for the problem instance at hand. */
3: while termination criteria do not hold do
4: selectedllh ActionSelection(str) /* Action Selection: Select a low-level
heuristic with the str-th action selection model (Section 3.2). */
5: stmp ApplyAction(scur , selectedllh ) /* Perturbation: Perturb the current
solution (scur ) with the selected low-level heuristic (selectedllh ). */
6: stmp ApplyLocalSearch(stmp ) /* Local Search: Apply a local search proce-
dure on the temporary solution stmp . */
7: scur AcceptanceCriterion(scur , stmp ) /* Acceptance: Accept which solution,
between scur , and stmp , will survive to the next iteration based on an Acceptance
criterion. */
8: CreditAssignment(selectedllh ) /* Credit Assignment: Score the used action
(selectedllh ) based on feedback from the problem at hand (Section 3.1).*/
9: end while
10: Return: the best solution found so far scur .
at hand. After having the initial solution, each iteration of HHILS operates ve
main steps. There is a set S which includes the k available perturbation low-level
search heuristics, S = {llh1 , llh2 , . . . , llhk }. HHILS exploits an action selection
method (ActionSelection(str)) to predict and select the most tting perturba-
tion low-level search heuristic included in S for the next step (line 4, see Sect. 3.2
for details). Having selected the perturbation low-level heuristic (selectedllh ), a
new solution, stmp , is generated by applying the selectedllh perturbation heuris-
tic to the current solution scur (line 5). After the perturbation, the new solution
(stmp ) is rened by a local search method, ApplyLocalSearch(stmp ) (line 6). The
local search procedure utilizes a set of greedy local search heuristics which are
applied in an iterative way. More specically, given a list L of the available
local search heuristics, L = {l1 , l2 , . . . , l }, at each repetition, a local search is
selected in a uniform random way to be applied to the current solution. If the
selected local search heuristic is not able to provide a better position for the
stmp , it is excluded from L, and another local search heuristic is selected. This
continues until an improved solution has been produced. By the end of the iter-
ative local search process, the result will be the incumbent solution (stmp ). In
the next step, HHILS will decide the best solution, between scur and stmp , to
use for the next iteration through the AcceptanceCriterion(scur , stmp ) proce-
dure (line 7). Here, we have adopted a Simulating Annealing acceptance rule
to allow worsening moves being accepted with a probability. The acceptance
probability can be calculated as p = e(f (scur )f (stmp ))/(T i ) , where f (scur ) and
f (stmp ) are the objective values of the incumbent (scur ) and temporary (stmp )
solutions, T is the temperature value with T R, (here is xed to T = 2) and
i is the mean improvement of the improving iterations [15]. The value essen-
tially normalizes the objective value dierence by a quantity that is not problem
dependent. In the last step, HHILS assigns a score to the utilized low-level per-
turbation heuristics (selectedllh ) involved based on its performance (incumbent
improvement) through the credit assignment module CreditAssignment
(selectedllh ) (line 8).
Four dierent perturbation strategies are used here, one from the mutation
and three from the ruin and recreate categories. The mutation heuristic (HM)
randomly un-assigns shifts based on an intensity parameter respecting the fea-
sibility of the solution. The three ruin and recreate heuristics (HR1HR3) are
all inspired by the one proposed in [20]. HR1 unassigns all shifts of random
employees from the schedule and recreates the schedule by prioritizing the objec-
tives related to weekdays and then to weekends. Then, greedy procedures are
used to satisfy the remaining objectives. A hill climbing procedure is employed
to improve the quality of the roster. HR1 destroys the solution by removing
the schedule of a medium number of employees. HR2 adopts a similar proce-
dure accepting a greater change to the solution, proportional to the number of
employees in the schedule, while HR3 slightly perturbs the solution by removing
the shifts from only one employee. Five dierent local searchers are also adopted
LS1LS5, The rst three are using dierent neighborhood operators from the lit-
erature, i.e., vertical , horizontal and new swaps respectively, while the last two
local searchers follow a variable depth search strategy with dierent neighbor-
hood operators (LS4: vertical and new, LS5: vertical, horizontal and new) [21].
A detailed description of all the employed low-level heuristics can be found in
the documentation of HyFlex [22].
3.1 Credit Assignment Module
In order to assess the quality of the last action performed each time, a credit
assignment module has been employed. The most conventional way to determine
the impact of each move and assign a credit to it, is to associate the search move
with the solution improvement caused by its application. To this end, we can
calculate the credit of an action based on the improvement of the incumbent
solution weighted by the eort paid to improve it.
More precisely, HHILS rewards each low-level perturbation heuristic accord-
ing to the ability to improve the incumbent solution normalized by the total time
spent to achieve this improvement. Let S = {llh1 , llh2 , . . . , llhk } be the set of k
available low-level perturbation heuristics and tllhi be the execution time con-
sumed by action llhi S to search the solution space. The total time consumed
by action llhi can be calculated according to tllhi = tllhi + tsp sp
llhi , where tllhi is the
execution time consumed by action llhi for the current iteration. The reward rllhi
of action llhi can, therefore, be calculated according to the following equation:
1+improvementllhi
rllhi = tllhi , where improvementllhi simply counts the number of
times action llhi improves the incumbent solution (i.e., f (scur ) < f (stmp ), where
scur is the incumbent solution and stmp is the solution produced by the search
operations at the current iteration). Notice that the value 1 in the numerator
is responsible for assigning non-zero rewards to actions that have not yet led
to an improvement of the incumbent. It also helps dene the minimal reward

for each action that is proportional to the time spent on searching for a new
(improved or not) solution.
In general, the estimation of the empirical quality of an action has to consider
also that recent rewards inuence the quality more than earlier. Thus, it is a
common practice to estimate the empirical quality of an action with a more
accurate and reliable way based on a simple moving average of the current and
past reward values [19,23,24]. As such, the empirical quality qllhi (t) of each action
(llhi ) in the current time step (t) is estimated in accordance with the following
relaxation mechanism:
qllhi (t + 1) = qllhi (t) + (rllhi (t) qllhi (t))

= (1 )qllhi (t) + rllhi (t) (1)
where (0, 1] is the adaptation rate which can amplify the inuence of the
most recent rewards over their history (here is xed to 0.1) [19,24].
Notice also, that the credit assignment module could employ any reward value
that can be measured during the search process to score the applied search oper-
ation. Representative examples of such rewards are the tness improvement [19],
ranking successful movements [19,25], and landscape analysis measures [26].
3.2 Action Selection Methodology

HHILS utilizes an action selection model to select the most suitable perturbation
low-level heuristic to apply during search. The proposed framework can adopt
any available action selection model. Many action selection models have been
proposed in the literature recently. They usually adopt theory and practical
algorithms motivated by dierent scientic elds, such as statistics, articial
intelligence, and machine learning. Some recent characteristic examples include
probability matching [24], adaptive pursuit [24], statistical based models [25],
and various reinforcement learning approaches [19,23,27].
Here, we employ six state-of-the-art action selection models with dierent
characteristics: the baseline model of uniform selection, proportional selection,
Probability Matching [24], Adaptive Pursuit [24], Soft-Max selection [23], and
the Upper Condence Bound Multi-Armed Bandit model [19,23]. For complete-
ness purposes, we provide a short description of the developed models. A full
description of all models can be found in the original publications [19,23,24].
The rst model, uniform selection (US) acts as a baseline model, since it selects
the available actions with equal probability regardless of their empirical quality,
i.e., through a uniform random distribution. It is worth noting that random-
ized models in either the action space or the parameter space of an algorithm
can be seen as essential baseline models [28]. Proportional selection (PS) simply
selects an action proportionally based on its empirical quality, thus the higher the
empirical quality value of an action the higher the probability of being selected.
Probability Matching (PM) and Adaptive Pursuit (AP) [24] are two well-
known and successful probabilistic schemes that update the selection probability
of an action by considering its empirical quality with respect to the remaining

actions. PM updates the probability of each action with respect to its empirical
quality while keeping a minimal probability for all actions to provide them with
the opportunity to be selected regardless of their eciency [24]. Similarly, AP
adopts a probabilistic scheme with a winner-takes-all strategy, in which only
ecient actions are promoted. AP discards the minimal probability value of each
action to clearly distinguish the ecient from the inecient actions, which in PM
are treated equally. Thus, instead of proportionally adapting the probabilities
of all available actions, it arises the probability of the best available action, and
reduces the probabilities of the remaining actions.
The nal two action selection models come from the eld of Reinforcement
Learning techniques, which have been successfully applied as action selection
models [19,23]. Specically, we employ two well known algorithms in this study,
the Softmax (SM) [23] and the Upper Confidence Bound (UCB1) Multi-armed
Bandit algorithm (MAB) [19,27,29]. The former essentially utilizes a Gibbs dis-
tribution to transform the empirical quality of each action to a probability.
The higher the empirical quality, the higher the probability of an action being
selected. Softmax also utilizes a temperature parameter to amplify or condense
dierences between the action probabilities. High temperature values lead to
probabilities which are almost equal for all available actions, whilst low tem-
perature values encourage larger dierences. The scheme becomes very greedy
towards selecting the best available action as the temperature value decreases
(tends to zero).
In the multi-armed bandit case the UCB1 algorithm deterministically selects
an action by following the principle of optimism in the face of uncertainty. The
principle acts according to an optimistic guess on the merit of the expected
empirical quality of each action and deterministically selects the action with
the highest guess. When the guess is not correct, the optimistic guess is being
rapidly decreased and the user is compelled to switch to a dierent action. If
the guess is correct then the user will exploit the associated action and incur
limited regret. Optimism can be formulated by an upper condence bound
that tries to balance the trade-o between exploration and exploitation of the
selection process amongst the available actions. Hence, UCB1 favors the selection
of the action that potentially exhibits the best reward (optimism in the face of
uncertainty), while providing the opportunity for scarcely tried actions to be
applied frequently. It is also worth mentioning that UCB1 is one of the very
successful multi-armed bandit techniques and achieves an optimal regret rate on
the multi-armed bandit formulation [29].
4 Experimental Results
We rstly present the experimental setup of this study (Sect. 4.1), which includes
details about the environment used, the considered problem instances, the pro-
posed as well as the state-of-the-art hyper-heuristics, and the parameter cong-
urations of all considered algorithms. We then proceed with the presentation of
the experimental results and thorough statistical analysis of the algorithms on

the given problem instances.
In this experimental study, we develop and evaluate eighteen dierent hyper-

heuristic approaches. Ten of them are based on the proposed HHILS frame-
work, while the remaining eight are state-of-the-art hyper-heuristics that have
been proposed in the literature. A family of hyper-heuristics can be dened
in the proposed HHILS by adopting dierent action selection models. In this
study, we develop the following six state-of-the-art action selection models
resulting in the following hyper-heuristics: ILS-US: The HHILS variant that
employs uniform selection among the available perturbation low-level heuris-
tics; ILS-PS: HHILS employing the proportional selection method (roulette
wheel); ILS-AP: HHILS employing the Adaptive Pursuit selection method [24];
ILS-MAB: HHILS employing the Upper Condence Bound Multi-Armed Ban-
dit method [19,23,27,29]; ILS-PM: HHILS employing the Probability Match-
ing selection method [24]; ILS-SM: HHILS employing the Soft-Max selection
method [23]. The HHILS variants select among four dierent low-level heuris-
tics (HM, HR1HR3). To evaluate the eect of the adaptive operator selection
procedure, we develop four dierent HHILS variants that instead of using the
proposed adaptive procedure, they adopt only one perturbation strategy. As
such, ILS-HM is the HHILS with the HM perturbation heuristic, ILS-HR1 is
the HHILS with the HR1 perturbation heuristic, and so on.
Additionally, we develop and compare the following eight state-of-the-art
hyper-heuristics, that are considered for the diversity of their characteristics and
their high performance: HH1, HH1A: The HH1 and HH1A (HH1adap) pre-
determined sequence non-worsening selection hyper-heuristic for nurse rostering
problems [30]; HH2, HH2A: The HH2 and HH1A (HH2adap) greedy absolute
largest improvement selection hyper-heuristic for nurse rostering problems [30];
PHunter: the Pearl Hunter hyper-heuristic [14]; SSHH: The sequence-based
selection hyper-heuristic with Hidden Markov Model [13]; FSILS: The Fair-
Share hyper-heuristic algorithm [15], which is a state-of-the-art methodology
in the eld; GIHH: The GIHH, or Adapt-HH, hyper-heuristic algorithm [16],
which was the winner of the CHESC 2011 competition [17]. Notice that all
hyper-heuristics have been implemented in JAVA using the HyFlex (v1.1) frame-
work [17,22,31,32].
In this study, we use a set of 39 real world nurse rostering problem instances
with dierent characteristics (Table 1 [10]1 ), where optimal solutions are known
for the majority of the instances. To acquire equitable comparisons across
all hyper-heuristics, we retain the same common parameters in all proposed
approaches. Additionally, we preserve the default parameters used in the com-
pared algorithms, as dened in the original works [1316,30].
1
More details can be found in http://www.cs.nott.ac.uk/tec/NRP/.
Table 1. Nurse rostering problem instances used.
BCV-1.8.1, BCV-1.8.2, BCV-1.8.3, BCV-1.8.4, BCV-2.46.1, BCV-2.46.1, BCV-3.46.2, BCV-4.13.1,

BCV-4.13.2, BCV-5.4.1, BCV-6.13.1, BCV-6.13.2, BCV-7.10.1, BCV-8.13.1, BCV-8.13.2,
BCV-A.12.1, BCV-A.12.2, ORTEC01, ORTEC02, GPost, GPost-B, QMC-1, QMC-2,
Ikeg-2Sh-DATA1, Ikeg-3Sh-DATA1, Ikeg-3Sh-DATA1.2, Valouxis-1, WHPP, LLR, Musa, Azaiez,
SINTEF, CHILD-A2, ERMGH-A, ERMGH-B, ERRVH-A, ERRVH-B, MER-A, QMC-A
To evaluate the performance of a hyper-heuristic on each problem instance,

we conducted 31 independent runs. For each of these runs, we considered the
best objective value gained as its performance value, where a lower objective
value denotes better performance. To facilitate comparisons among the consid-
ered hyper-heuristics we utilize the following metrics: (a) the normalized objec-
tive value f (), to fair compare the results across all problem instances, (b) the
regret of a hyper-heuristic reg, to compare the considered algorithm with the
best performing one, and (c) the davg metric to measure the distance of the aver-
age performance of a hyper-heuristic from the best known solution for a given
instance.
We normalize the objective values in a common range of values by applying
a linear transformation of all the objective values gained, S = [Smin , Smax ], to a
normalized range T = [Tmin , Tmax ], here T = [0, 1], where Smin is the best known
solution of the considered problem and Smax is maximum objective function
value observed by all utilized hyper-heuristics. The linear transformation can be
calculated according to function f : S T, f (y) = (y Smin )/(Smax Smin ),
where y S R is the performance value obtained by a hyper-heuristic on a
given problem instance. For a given algorithm A from a set of n algorithms to
compare A A = {Aj , j = 1, . . . , n}, and a specic execution run i, we denote
p
the regret of A as regA (i) = fAp (i) mini,Aj A (fAp ), where p is the considered
p
problem instance, fA (i) is the tness value that has been attained for A at
the i-th run for the problem instance p and mini,Aj A (fAp ) is the lowest tness
value across all execution runs of the algorithms included in A. Consequently, reg
denotes the regret preferring an algorithm A to the best performing algorithm for
a specic problem instance. Equivalently, davg is determined as davg = BKS BKS ,
where BKS is the value of the best known solution according to the literature,
regardless of whether the solution is optimal or not, and denotes the average
performance value of the utilized algorithm for all executed runs on a problem
instance. Thus, declares the normalized distance of the average performance
of an algorithm from the BKS for a specic instance.
To facilitate fair comparisons, for each hyper-heuristic, each problem instance
and each execution run, we employ the same execution time budget, i.e. 10 min
of CPU time as measured by the benchmarking program provided in HyFlex [17,
22]. All experiments required have been conducted on a high performance cluster
that has computation nodes with Intel Xeon E5645 CPUs and 24 GB of RAM
running GNU/Linux Operating System. For this hardware, the allowed time
limit equals to 646 s (tallowed = 646 s).
4.2 Experimental Results and Analysis

Table 2 exhibits summarizing performance statistics for the developed hyper-
heuristics on all considered nurse rostering problem instances, in terms of the
normalized objective (f ), the regret (reg) and the davg metrics used. Specically,
for each algorithm and each metric, the mean (X ), median (mX ) and standard
deviation (X ) values are presented, where X {f, reg, davg }. We divide the
table in three categories, the HHILS variants using single perturbation strategies,
the proposed adaptive HHILS variants and the state-of-the-art hyper-heuristics.
To improve the presentation of the results, we highlight with boldface font the
cases where either the mean or the median metric values indicate best perfor-
mance (i.e., the smallest values) across each category of algorithms.
Table 2. Summarizing statistics of the normalized objective values, regret, and davg
metrics of all hyper-heuristics across all considered problem instances.
Algorithm f mf f reg mreg reg davg mdavg davg

ILS-HM 0.1037 0.0162 0.2008 0.0690 0.0040 0.1424 0.0740 0.1670 0.0000
ILS-HR1 0.2758 0.1907 0.2986 0.2497 0.1591 0.2785 0.6253 0.4376 0.9592
ILS-HR2 0.1091 0.0203 0.1993 0.0724 0.0058 0.1433 0.0680 0.1369 0.0192
ILS-HR3 0.1110 0.0181 0.2037 0.0757 0.0066 0.1469 0.0735 0.1456 0.0221
ILS-AP 0.1053 0.0168 0.2016 0.0692 0.0047 0.1438 0.0632 0.1482 0.0135
ILS-MAB 0.1083 0.0185 0.2017 0.0731 0.0069 0.1441 0.0695 0.1410 0.0187
ILS-PM 0.1103 0.0190 0.2025 0.0747 0.0080 0.1451 0.0764 0.1557 0.0115
ILS-PS 0.1065 0.0185 0.2000 0.0713 0.0075 0.1414 0.0692 0.1363 0.0112
ILS-SM 0.1102 0.0196 0.2014 0.0739 0.0073 0.1443 0.0696 0.1458 0.0166
ILS-US 0.1104 0.0199 0.2026 0.0747 0.0084 0.1461 0.0765 0.1435 0.0157
FSILS 0.1114 0.0242 0.1843 0.0776 0.0150 0.1139 0.0904 0.1451 0.0276
GIHH 0.1286 0.0231 0.2070 0.0940 0.0114 0.1524 0.1258 0.1738 0.0545
HH1 0.1409 0.0349 0.2048 0.1087 0.0227 0.1489 0.1622 0.2071 0.0898
HH1A 0.1358 0.0322 0.2005 0.1040 0.0227 0.1409 0.1523 0.1977 0.0755
HH2 0.1334 0.0227 0.2035 0.0994 0.0108 0.1468 0.1321 0.1941 0.0508
HH2A 0.2866 0.1364 0.3151 0.2613 0.1250 0.2921 0.4349 0.4449 0.2222
PHunter 0.1332 0.0219 0.2285 0.1000 0.0122 0.1816 0.1409 0.2409 0.0726
SSHH 0.1320 0.0242 0.2139 0.0992 0.0144 0.1620 0.1467 0.2115 0.0726
Considering the HHILS variants with a single perturbation strategy, Table 2

clearly suggests that the best performing hyper-heuristics are ILS-HM and
ILS-HR2, in terms of mean and median normalized objective values. Notice
also that ILS-HM is the best performing algorithm comparing against all con-
sidered hyper-heuristics. The remaining two cases exhibit large dierences in
both mean and median normalized objective values, which indicates that there
is a large dierence in the eectiveness of the low-level perturbation strate-

gies. This behavior clearly motivates the proposed approach that endeavors to
learn and identify the best performing perturbation strategy from the available
ones. Regarding the proposed HHILS variants that employ the adaptive oper-
ator selection mechanism, ILS-AP and ILS-PS exhibit the best performance in
terms of the mean and median normalized objective values among the HHILS
variants, respectively. ILS-PM, ILS-SM and ILS-MAB behave similarly with ILS-
US, i.e., with the uniform selection strategy, which indicates that the adaptive
schemes are not capable of identifying and learning the best performing pertur-
bation strategy. However, it can be observed that even a random selection of the
available choices is eective in this set of problem instances. In addition, it is
worth mentioning that all adaptive HHILS variants outperform all state-of-the-
art hyper-heuristics in terms of mean normalized objective values. Among the
state-of-the-art hyper-heuristics the most promising algorithm is FSILS. Notice
that FSILS follows a similar iterated local search based hyper-heuristic that does
not incorporate any learning, or action selection mechanism, to identify the best
perturbation strategy. HHILS uses a dierent local search procedure (it does not
employ an active list of local searchers) than FSILS and does not incorporate any
restarting mechanism [15]. The good performance of FSILS indicates that the
ILS paradigm is eective in the considered problem instances. The signicantly
better performance of HHILS adaptive variants also denotes that the combina-
tion of dierent perturbation operators and the learning mechanism enhance the
search process. Similar observations can be made for the remaining robustness
metrics, where they suggest that selecting any of the HHILS variants will not
lead to higher regret (or davg ) than using the state-of-the-art hyper-heuristics.
To further understand the behavior of the adaptive HHILS variants, Fig. 1
illustrates the selection frequency of the available perturbation strategies across
all problem instances, averaged over all independent runs. It is clear that ILS-AP
and ILS-PS show the largest frequency variation among the available strategies.
In contrast, ILS-MAB, ILS-SM and ILS-PM tend to equally select all available
strategies, making them behave similarly with the ILS-US (uniform selection),
which validates the observed performance values in Table 2.
The application of the Friedman rank sum test [33] clearly suggests that sta-
tistically signicant dierences in performance occur (p < 0.0001, 2 = 244.8393
at the 5% signicance level) among the considered hyper-heuristics across all
problem instances. Therefore, we carry out a post-hoc analysis by perform-
ing pairwise Wilcoxon-signed rank tests between any two performance samples
obtained by the considered hyper-heuristics. The tests suggest that all proposed
HHILS adaptive variants exhibit signicant dierences in performance against
all state-of-the-art hyper-heuristics. ILS-AP signicantly outperforms all HHILS
variants apart from ILS-PS, which behaves equally well. ILS-MAB, ILS-PM,
ILS-SM behave equally well with ILS-US. Considering the HHILS variants with
single perturbation strategy ILS-HM outperforms all algorithms, while ILS-HR2
behaves equally well with the majority of the HHILS with adaptive strategies,
Azaiez BCV1.8.1 BCV1.8.2 BCV1.8.3 BCV1.8.4 BCV2.46.1

100%
75%
50%
25%
0%
BCV3.46.1 BCV3.46.2 BCV4.13.1 BCV4.13.2 BCV5.4.1 BCV6.13.1
100%
75%
50%
25%
0%
BCV6.13.2 BCV7.10.1 BCV8.13.1 BCV8.13.2 BCVA.12.1 BCVA.12.2
100%
75%
50%
25%
0%
CHILDA2 ERMGHA ERMGHB ERRVHA ERRVHB GPost
100%
75%
50%
25%
0%
GPostB Ikeg2ShDATA1 Ikeg3ShDATA1 Ikeg3ShDATA1.2 LLR MERA
100%
75%
50%
25%
0%
Musa ORTEC01 ORTEC02 QMC1 QMC2 QMCA
100%
75%
50%
25%
0%
SINTEF Valouxis1 WHPP P B M S M P B M S M P B M S M
A MA P P S SA MA SP SP SS SA MA SP SP SS
100% ILS ILS ILS ILS ILS IL ILS IL IL IL IL ILS IL IL IL
75%
50%
25%
0% HM HR1 HR2 HR3
P B M S M P B M S M P B M S M
A MA P P S SA MA SP SP SS SA MA SP SP SS
ILS ILS ILS ILS ILS IL ILS IL IL IL IL ILS IL IL IL
Fig. 1. Frequency graphs of the selected strategies by the adaptive HHILS variants for
all problem instances averaged over all simulations
apart from ILS-AP which is superior. From the state-of-the-art hyper-heuristics

GIHH, PHunter and SSHH behave similarly, while FSILS outperforms them.
To facilitate further comparisons, Table 3 exhibits descriptive statistics in
terms of the best objective values attained by the considered hyper-heuristics
among the majority of the tested problem instances. Notice that, we do not
present results for BCV-5.4.1, BCV-8.13.1, BCV-8.13.2, LLR, Musa, Azaiez
instances, since all hyper-heuristics solved these instances to the best-known
solution without any deviation. The presented results align well with the afore-
mentioned observations, while also suggest that there are instances where a large
deviation in the observed performance among the hyper-heuristics exists (e.g. in
CHILD-A2, ERRVH-A, ERRVH-B, and MER-A). This might also indicate that
these problems are more challenging than others.
Table 3. Summarizing statistics of the objective values for all hyper-heuristics across all problem instances on B1 set.
An Iterated Local Search Framework with Adaptive Operator Selection
105
5 Conclusions
In this study, we proposed a simple and eective Iterated Local Search based
selection hyper-heuristic framework that adopts the adaptive operator selec-
tion paradigm to successfully address a wide variety of nurse rostering problem
instances. It employs an action selection model to select dierent perturbation
strategies and a credit assignment module to appropriately score them. The pro-
posed framework is able to adopt any action selection model and credit assign-
ment mechanism available in the literature. In this study, we have tested six
dierent action selection models resulting in new competitive hyper-heuristics.
The high level nature of the framework makes it widely applicable to new or
unseen problem instances/domains without requiring further modications.
The adaptive characteristics of the proposed framework are investigated by
comparing with its non-adaptive variants, while its performance is evaluated
through comparisons with 8 state-of-the-art hyper-heuristics on 39 dierent
nurse rostering problem instances. The experimental results suggest that the pro-
posed framework operates signicantly better against the state-of-the-art hyper-
heuristics. The proposed adaptive mechanisms seem to be eective across the
majority of the problem instances, with the Adaptive Pursuit and the simple
proportional action selection model to be able to learn and identify the most
promising perturbation strategies. The remaining three considered action selec-
tion models operate similarly with the uniform selection, which indicates that
they are not able to identify the best performing perturbation strategy. However,
even a simple random selection performs signicantly better than the majority
of the state-of-the-art algorithms. Therefore, further experimentation and analy-
sis of the adaptive strategies on more nurse rostering problem instances have to
be performed to draw safe conclusions about their behavior. Future work will
also include comparisons with specialized state-of-the-art heuristics developed
for nurse rostering problems.
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Learning a Reactive Restart Strategy
to Improve Stochastic Search
Serdar Kadioglu1 , Meinolf Sellmann2 , and Markus Wagner3(B)

1
Department of Computer Science, Brown University, Providence, RI, USA
serdark@cs.brown.edu
2
Cortlandt Manor, Cortlandt, NY, USA
meinolf@gmail.com
3
Optimisation and Logistics, The University of Adelaide, Adelaide, SA, Australia
markus.wagner@adelaide.edu.au
Abstract. Building on the recent success of bet-and-run approaches for

restarted local search solvers, we introduce the idea of learning online
adaptive restart strategies. Universal restart strategies deploy a xed
schedule that runs with utter disregard of the characteristics that each
individual run exhibits. Whether a run looks promising or abysmal, it
gets run exactly until the predetermined limit is reached. Bet-and-run
strategies are at least slightly less ignorant as they decide which trial
to use for a long run based on the performance achieved so far. We
introduce the idea of learning fully adaptive restart strategies for black-
box solvers, whereby the learning is performed by a parameter tuner.
Numerical results show that adaptive strategies can be learned eectively
and that these signicantly outperform bet-and-run strategies.
Keywords: Restart strategies Adaptive methods Parameter tuning
1 Introduction
Restarted search has become an integral part of combinatorial search algorithms.
Even before heavy-tailed runtime distributions were found to explain the massive
variance in search performance [1], in local search restarts were commonly used
as a search diversication technique [2].
Fixed-schedule restart strategies were studied theoretically in [3]. For SAT
and constraint programming solvers, practical studies followed. For example,
one study found that there is hardly any dierence between theoretically opti-
mal schedules and simple geometrically growing limits [4]. SAT solvers used
geometrically growing limits for quite some time before the community largely
adapted theoretically optimal schedules (whereby the optimality guarantees are
based on assumptions that actually do not hold for clause-learning solvers where
consecutive restarts are not independent). Audemard and Simon [5] argued that
xed schedules are suboptimal for SAT solvers and designed adaptive restarts
strategies for one SAT solver specically.
https://doi.org/10.1007/978-3-319-69404-7_8
110 S. Kadioglu et al.
In this paper, we describe a general methodology for embedding any black-

box optimization solver into an adaptive stochastic restart framework. The
framework monitors certain key performance metrics that are based on the evo-
lution of the objective function values of the solutions found. Based on these
observations, the method then adaptively computes scores that aect the like-
lihood whether we continue the current run beyond the original limit, whether
we start a new run, or whether we continue the best run so far. We employ an
automatic parameter tuner to learn how to adapt these probabilities dependent
on the observed performance metrics.
In the following, we recap the idea of bet-and-run strategies. We continue
with reviewing the recently introduced idea of hyper-parameterizing local search
solvers to achieve superior online adaptive behavior. We then introduce the idea
of using automatic hyper-reactive search tuning for learning adaptive restart
strategies. Finally, we present experimental results that clearly show that adap-
tive search signicantly outperforms bet-and-run strategies.
2 Restart Strategies
Nowadays, stochastic search algorithms and randomized search heuristics are
frequently restarted: If a run does not conclude within a pre-determined limit,
we restart the algorithm. This was shown to to help avoid heavy-tailed runtime
distributions [1]. Due to the added complexity of designing an appropriate restart
strategy for a given target algorithm, the two most common techniques used are
to either restarts with a certain probability at the end of each iteration, or to
employ a xed schedule of restarts.
Some theoretical results exist on how to construct optimal restart strategies.
For example, Luby et al. [3] showed that, for Las Vegas algorithms with known
run time distribution, there is an optimal stopping time in order to minimize the
expected running time. They also showed that, if the distribution is unknown,
there is a universal sequence of running times which is the optimal restarting
strategy up to constant factors.
Fewer results are known for the optimization case. Marti [6] and Lourenco
et al. [7] present practical approaches, and a recent theoretical result is presented
by Schoenauer et al. [8]. Particularly for the satisability problem, several studies
make an empirical comparison of a number of restart policies [9,10].
Quite often, classical optimization algorithms are deterministic and thus can-
not be improved by restarts. This also appears to hold for certain popular modern
solvers, such as IBM ILOG CPLEX. However, characteristics can change when
memory constraints or parallel computations are encountered. This was the ini-
tial idea of Lalla-Ruiz and Vo [11], who investigated dierent mathematical
programming formulations to provide dierent starting points for the solver.
Many other modern optimization algorithms, while also working mostly deter-
ministically, have some randomized component, for example by choosing a random
starting point. Two very typical uses for an algorithm with time budget t are to
(a) use all of time t for a single run of the algorithm (single-run strategy), or (b) to
Learning a Reactive Restart Strategy to Improve Stochastic Search 111
make a number of k runs of the algorithm, each with running time t/k (multi-run
strategy).
Extending these two classical strategies, Fischetti et al. [12] investigated the
use of the following Bet-and-Run strategy with a total time limit t:
Phase 1 performs k runs of the algorithm for some (short) time limit t1 with
t1 t/k.
Phase 2 uses remaining time t2 = t k t1 to continue only the best run from
the rst phase until timeout.
Note that the multi-run strategy of restarting from scratch k times is a special
case by choosing t1 = t/k and t2 = 0 and the single-run strategy corresponds to
k = 1; thus, it suces to consider dierent parameter settings of the bet-and-run
strategy to also cover these two strategies.
Fischetti et al. [12] experimentally studied such a Bet-and-Run strategy for
mixed-integer programming. They explicitly introduce diversity in the starting
conditions of the used MIP solver (IBM ILOG CPLEX) by directly accessing
internal mechanisms. In their experiments, k = 5 performed best.
Recently, Friedrich et al. [13] investigated a comprehensive range of Bet-
and-Run strategies on the traveling salesperson problem and the minimum ver-
tex cover problem. Their best strategy was Restarts40 1% , which in the rst phase
does 40 short runs with a time limit that is 1% of the total time budget and
then uses the remaining 60% of the total time budget to continue the best run
of the rst phase. They investigated the use of the universal sequence of Luby
et al. [3] as well, using various choices of t1 , however, it turned out inferior.
The theoretical analysis is provided by Lissovoi et al. [14], who investigated
Bet-and-Run for a family of pseudo-Boolean functions, consisting of a plateau
and a slope, as an abstraction of real tness landscapes with promising and
deceptive regions. The authors showed that Bet-and-Run with non-trivial k
and t1 are necessary to nd the global optimum eciently. Also, they showed that
the choice of t1 is linked to properties of the function, and they provided a xed
budget analysis to guide selection of the bet-and-run parameters to maximise
expected tness after t = k t1 + t2 tness evaluations.
The goal of our present research is to address the two challenges encountered
in previous works: the need to set k and t1 in case of Bet-and-Run, and the
general issue of inexibility in previous approaches. Our framework can decide
online whether (i) the current run should be continued, (ii) the best run so far
should be continued, or (iii) a completely new run should be started.
3 Learning Dynamic Parameter Updates
Our objective is to provide a generic framework for making restart strategies

adaptive for any optimization solver. To this end we build on the idea to use
parameter tuners for training adaptive search strategies [15,16] and a recently
proposed approach for constructing a hyper-reactive dialectic search solver [17].
In [17], an existing local search meta-heuristic called dialectic search [18]

was modied in such a way that the search decisions (when and how much to
diversify, how strongly to intensify, when to restart, etc.) were taken with regard
to the way how the optimization was observed to progress. In essence, the solver
tracked features of the optimization process itself and then tied these to decisions
(such as: which percentage of variables to modify to generate a new start point)
via logistic regression functions. The weights of these functions, one for each
meta-heuristic search decision, then became the hyper-parameters of the solver.
Key to making this work in practice is an eective method for learning the
weights in the logistic regression functions. Since the only immediately mean-
ingful observation is the overall performance of the solver, parameter tuner
GGA [19] was used to learn which weights result in good performance.
4 A Hyper-Parameterized Restart Strategy

We now combine the two core ideas presented above. Namely, the idea of consid-
ering a batch of runs with the option to continue some of them, and the idea to
automatically learn which run to continue or whether to start a new run based
on the observed performance characteristics of past runs.
The rst ingredient we need are features that somehow give us an idea of the
big picture of what is going on when tackling the instance at hand.
4.1 Features
Whenever a restart decision has to be made, we have three options. We can

either continue the current run, we can continue the best run so far, or we can
start a completely new run. For each of these options we essentially track two
values: The rst tells us how good each run looked initially, the second what the
trajectory looks like for making further progress.
For the current run and the best run so far, we record their best objective
function found after the initial limit. For the new run option, we track how well
any new run did after the initial limit and compute the running average.
For the trajectory of the current and the best run, we extrapolate the perfor-
mance improvement achieved between the best solution found in the initial run
and the best performance achieved so far. The extrapolation point is the end of
the remaining time we have for the optimization.
For the new runs, to get an estimate how well we might do if (from now
until the overall time limit is reached) all we did was run new runs, we consider
the standard deviation in objective function performance. Then, we estimate the
trajectory as the average minus the standard deviation times the square root of
two times the logarithm of the number of new runs we can still aord to conduct.
While not exact, this is a lower bound for the minimum of repeated stochastic
experiments for which all we know are the mean and the standard deviation [20].
We thus compute six data points that we can use to decide whether to con-
tinue the current run, give the current best run more time, or start a completely
new run. One complication arises. Namely, for dierent instances, the objec-
tive function values observed will generally operate on vastly dierent scales.
However, to learn strategies oine, we need to compute weights, and these need
to work with all kinds of instances. Consequently, rather than taking average
and projected objective function values at face value, we rst normalize them.
In particular, we consider the three initial values (best found in initial time
interval for current and best, and running average of best found for all new runs)
and normalize them between 0 and 1. That is, we shift and scale these values in
such a way that their smallest will be 0, the largest will be 1, and the last will be
somewhere between 0 and 1. Analogously, we normalize the trajectory values.
On top of the six features thus computed we also use the percentage of
overall time that has already elapsed, the percentage of overall time aorded in
the beginning where all we do is restart a new run every time, and the time a
new run will be given as percentage of total time left. In total we thus arrive at
nine features.
4.2 Turning Features into Scores
Now, to compute the score for each of the three possibilities (continue current
run, continue best run so far, and start a new run) we compute the following
function
1
pk (f ) ,
1 + exp(w0k + i fi wik )
whereby k {1, 2, 3} marks whether the function marks the score for continuing
the current run, continuing the best run, or starting a new run, and f R9
is the feature vector that characterizes our search experience so far. Note that
pk (f ) (0, 1), whereby the function approaches 0 when the weighted sum in
the denominators exponential function goes to innity, and how the function
approaches 1 when the same sum approaches minus innity. Finally, note that
we require a total of 30 weights to dene the three functions. These weights will
be learned later by a parameter tuner to achieve superior runtime behavior.
4.3 The Reactive Restart Framework
Given the weights wik with k {1, 2, 3} and i {0, . . . , 9}, we can now dene
the framework within which we can embed any black-box optimization solver.1
1
We say black-box because we do not need to know anything about the inner workings
of the solver. However, we make two assumptions. First, that we can set a time limit
to the solver where it stops, and that we can add more time and continue the
interrupted computation later. Second, that whenever the solver stops it returns
information when it found the rst solution, when it found the best solution so far,
and what the quality of the best solution found so far is.
Algorithm 1. Reactive Restart Framework Algorithm

k{1,2,3}
1: function ReactiveRestarts (S,x,timeout,k,r,wi{0,...,9} )
2: (initTime, best) S(x, newRun, stopAtF irstSolution)
3: interval r initTime
4: b S(x, continueLastRun, interval initTime)
5: elapsedTime interval
6: Update(best, b)
7: while elapsedTime k timeout do
8: (a, b) S(x, interval)
9: elapsedTime elapsedTime + interval
10: Update(initTime, best, a, b)
12: Init(F )
13: while elapsedTime timeout do
14: pk 1+exp(wk + 1

wk F )
k {1, 2, 3}
0 i i i
k
15: pk p1 +pp2 +p3 k {1, 2, 3}
16: pick random x [0, 1]
17: if x p1 then
18: b S(x, continueBestRun, interval)
20: Update(best, b)
21: else if x p1 + p2 then
22: b S(x, continueLastRun, interval)
24: Update(best, b)
25: else
26: (a, b) S(x, newRun, interval)
28: Update(initTime, best, a, b)
30: Update(F )
31: returnbest
We present a stylized version of our framework in Algorithm 1. Given are

a randomized optimization algorithm S, an input x, a timeout, a fraction k
[0, 1], a factor r 1, and weights w. The framework rst runs S on x until a
rst solution is computed. It records the time to nd this solution and sets the
incremental time interval each run is given to r times this input-dependent value.
Next, S run on x is continued until this incremental time interval is reached.
The best solution seen so far is recorded.
Now, the rst phase begins, which lasts for the fraction of the total time
allowed as specied by k. In this rst phase, we start a new run on x every single
time, whereby we update the best solution seen so far and the time it takes each
time to nd a rst solution. The function Update is assumed to record the best
solution quality found so far as well as to maintain the running average of the
time it took to compute a rst solution for each new run.
After the rst phase ends, we initialize the features based on the search
experience so far. Then, we enter the main phase. Based on the given weights and
the current features we compute scores for the three options how we can continue
the computation at each step. We then choose randomly and proportionally to
these scores whether we continue the best run so far, the last run, or whether
we begin a new run.
No matter which choice we always keep the best solution found so far up to
date. When we choose to start a new run, we also update the running average of
the times it takes to nd a rst solution as well as the incremental time interval
that results from this running average times the factor r. Finally, we update the
features and continue until the time has run out.
The last ingredient needed to apply this framework in practice is a method for
learning the weights w. Based on a training set of instances, we compute weights
that result in superior performance using the gender-based genetic algorithm tuner
GGA [19], following the same general approach for tuning hyper-parameterized
search methods as introduced in [17].
5 Experimental Analysis
We now present our numerical analysis. First, we briey introduce the combina-
torial optimization problems, the solvers, and the instances used in our experi-
ments. Second, we describe our comprehensive data collection, which allows us
to conduct our investigations completely oine, that is, without the need of
running any additional experiments. Third, we present the results of our inves-
tigations which show the eectiveness of our online method.
5.1 Problems and Benchmarks

First, we briey introduce the two considered NP-complete problems, as well as
the corresponding solvers and benchmarks used in our investigations.
Traveling Salesperson. The Traveling Salesperson Problem (TSP) considers

an edge-weighted graph G = (V, E, w), the vertices V = {1, . . . , n} are referred to
n1
as cities. It asks for a permutation of V such that i=1 w((i), (i + 1)) +
w((n), (1)) (the cost of visiting the cities in the order of the permutation and
then returning to the origin (1)) is minimized.
Natural applications of the TSP are in areas like planning and logistics [21],
but they are also encountered in a large number of other domains, such as genome
sequencing, drilling problems, aiming telescopes, and data clustering [22]. TSP
is one of the most important (and most studied) optimization problems.
We use the Chained-Lin-Kernighan (Linkern) heuristic [23,24], a state-of-
the-art incomplete solver for the Traveling Salesperson problem. Its stochastic
behavior comes from random components during the creation of the initial tour.
The TSPlib is a classic repository of TSP instances [25]. For our investi-
gations, we pick all 112 instances from TSPlib, and as additional challenging
instances ch71009, mona-lisa100k, and usa115475.
Minimum Vertex Cover. Finding a minimum vertex cover of a graph is a

classical NP-hard problem. Given an unweighted, undirected graph G = (V, E),
a vertex cover is dened as a subset of the vertices S V , such that every edge
of G has an endpoint in S, i.e. for all edges {u, v} E, u S or v S. The
decision problem k-vertex cover decides whether a vertex cover of size k exists.
We consider the optimization variant to nd a vertex cover of minimum size.
Applications arise in numerous areas such as network security, scheduling and
VLSI design [26]. The vertex cover problem is also closely related to the problem
of nding a maximum clique. This has a range of applications in bioinformatics
and biology, such as identifying related protein sequences [27].
Numerous algorithms have been proposed for solving the vertex cover prob-
lem. We choose FastVC [28] over the popular NuMVC [29] as a solver for the
minimum vertex cover problem as it works better for massive graphs. FastVC is
based on two low-complexity heuristics, one for initial construction of a vertex
cover, and one to choose the vertex to be removed in each exchanging step, which
involves random draws from a set of candidates.
For our experimental investigations, we select all 86 instances used in [28].
Among these, the number of vertices ranges from about 1000 to over 4 million,
and the number of edges ranges from about 2000 to over 56 million.
5.2 Data Collection

We recorded 10,000 independent, regular runs of the original solvers on each of
the 115 TSP instances and on each of the 86 MVC instances. For TSP, the time
limit per instance was 1 h. For MVC, we allowed 100 s. The runs were conducted
on a compute cluster with Intel Xeon E5620 CPUs (2.4 GHz).
For each run, we make a record whenever a solver nds a better solution,
together with the solution quality. Altogether, the records of our 20,000 runs
take up over 8 GB when GZ-compressed with default settings. We plan to make
these les publicly available (upon nding a suitable webserver) as a resource
for studying the behaviour of these algorithms.
5.3 Training of Hyper

For each of the two benchmarks, we used two thirds of the respectively available
instances for training. That is, we handed the parameterized framework to a
recently improved version of GGA that uses surrogate models to predict where
improved parameterizations may be found [30] We ran GGA for 70 generations
with a population size of 100 individuals. The random replacement rate was set
to 5%, the mutation rate was set to 5% as well.
5.4 Results
Following the training of Hyper on two thirds of the instances (per problem
domain), we are left with 38 of the 115 TSP instances and 28 of the 86 MVC
instances. We use these to compare the performance of the following investigated
approaches:
1. Single: the solver is run once with a random seed, allowing it to run for the
total time given;
2. Restarts: the solver is restarted from scratch whenever a preset time limit
is reached, and this loop is repeated until time is up;
3. Luby: restarts based on the xed Luby sequence [3], where one Luby time
unit is based on ve times the time the rst run needs to produce the rst
solution;
4. Bet-and-Run: the previously described bet-and-run strategy by Friedrich
et al. [13];
5. Hyper: our trained hyper-parameterized bet-and-run restart strategy, as
described above.
We will analyze the outcomes using several criteria. First, we compare the
performance gaps achieved with respect to the optimal solution possible within
the time budget.2 Second, we consider the number of times an approach is able
to nd the best possible solution. Third, we compare the amount of time needed
in order to compute the nal results.
To start o, Tables 1 and 2 show the results of the individual solvers across
the sets of 38 and 28 instances. Note that we are using the problem domain
names TSP and MVC instead of the respective solvers to facilitate reading.
We observe that the number of times the best possible solution is found
increases with increasing time budget. Note that this is not natural as the best
possible solution is the best possible solution for the respective time limit! The
fact that the relative gap decreases anyhow is therefore a reection of the fact
that the best restart can actually nd the best solution rather quickly. With
increasing time limits, the restarted approaches thus have more buer to nd
this best quality solution as well.
Next, we nd that Single and Restarts are clearly outperformed by the
other three approaches across both problem domains and across all total time
budgets. On TSP, Hyper achieves less than half the performance gap of Bet-
and-Run when the total time budget is only 100 s. This advantage for Hyper
becomes more and pronounced as the budget increases to 5,000 s. For this time
limit, Hyper has a six-times lower average gap than Bet-and-Run, which
is marked improvement. At the same time, Bet-and-Run can nd the best
solutions in only 67% of the runs, whereas Hypers success rate is 84%.
MVC can be seen as a little bit more challenging in our setting, as the com-
putation time budgets were rather short and FastVC encountered signicant
initialization times on some of the large instances. As a consequence, the number
of times where no solution has been produced by the various approaches is higher
than for TSP, however, this number decreases with increasing time budget.
On MVC, Hyper and Bet-and-Run are really close in terms of average per-
formance gap, however, there is an advantage for Hyper in number of times the
best possible solutions are found. In practice this is still a substantial improve-
ment.
2
This best possible solution is the best solution provided within the given time limit
by any of the 10,000 runs we conducted.
Table 1. TSP results. Shown are time in seconds, and performance gap from the best
possible solution within the respective time limit. solutions and no solutions refer
to the number of times the approach has produced any solution at all. best found lists
the number of times the best possible solution was found given 380 runs (38 instances
10 independent runs). Highlighted in dark blue and light blue are the best and second
best average approaches.
Single
time average average
solutions no solutions best found
budget performance time
100 380 0 234 0.1415 12
200 378 2 239 0.1368 21
500 380 0 266 0.0885 95
1000 380 0 266 0.0877 105
2000 380 0 266 0.0762 165
5000 380 0 266 0.0596 290
Restarts
100 380 0 252 0.0689 21
200 380 0 255 0.0618 35
500 380 0 259 0.0519 61
1000 380 0 261 0.0474 98
2000 380 0 261 0.0457 154
5000 380 0 258 0.0435 268
Luby
100 380 0 296 0.0274 19
200 380 0 299 0.0189 32
500 380 0 309 0.0135 75
1000 380 0 317 0.0108 127
2000 380 0 318 0.0090 229
5000 380 0 322 0.0070 476
Bet-and-Run
100 380 0 244 0.0487 5
200 380 0 245 0.0473 6
500 380 0 246 0.0444 8
1000 380 0 248 0.0436 13
2000 380 0 251 0.0429 22
5000 380 0 256 0.0419 49
Hyper
100 380 0 295 0.0216 15
200 380 0 302 0.0142 26
500 380 0 307 0.0132 57
1000 380 0 307 0.0090 87
2000 380 0 319 0.0077 178
5000 380 0 321 0.0066 322
Hyper vs Single
16 15 12 12 12 12
TSP:
(gap) 25 25 25 25
22 22
Hyper vs Restarts
14 13 13 13 13 13
TSP:
2 2
(gap) 22 23 24 24 24 24
Hyper vs Luby
TSP: 27 3 29 29 29 29
2 29 2
(gap) 9 5 8 8 8 7
Hyper vs Bet-and-Run
13 14 13 13 14 13
TSP:
3 2 3 2 2
(gap) 22 22 22 23 23
23
time budget: 100s 200s 500s 1000s 2000s 5000s

Hyper vs Single
9 13 11 12
MVC:
7 2
(gap) 7
9 6
11 7 7 8
Hyper vs Restarts
6 11 11 11
MVC: 8
3
(gap) 7
9 8 5
13 8 9
Hyper vs Luby
8 7 8 7
MVC: 4 7 8 3
(gap) 10
15 13 11 8
Hyper vs Bet-and-Run
8 8 11
MVC: 6
2 3
(gap) 19 21 18 14
time budget: 50s 100s 200s 500s
Fig. 1. Statistical comparison of Hyper with the other approaches using the Wilcoxon
rank-sum test (signicance level p = 0.05). The approaches are compared based on the
quality gap to the best possible solution (smaller is better).
The colors have the following meaning: Green indicates that Hyper is statistically
better, Red indicates that Hyper is statistically worse, Light gray indicates that both
performed identical, Dark gray indicates that the dierences were statistically insignif-
icant. We have chosen pie charts on purpose because they allow for a quick qualitative
comparison of results. (Color gure online)
Table 2. MVC results. Shown are time in seconds, and performance gap from the best
possible solution within the respective time limit. solutions and no solutions refer
to the number of times the approach has produced any solution at all. best found
lists the number of times the respective best possible solution has been found given
280 runs (28 instances 10 independent runs). Highlighted in dark blue and light blue
are the best and second best average approaches.
Single
5 211 69 74 0.1097 3
10 223 57 76 0.4558 5
20 254 26 80 0.6181 9
50 264 16 98 0.2273 19
Restarts
5 228 52 76 0.1111 3
10 252 28 82 0.4140 6
20 268 12 88 0.6128 12
50 278 2 101 0.1802 25
Luby
5 228 52 80 0.1064 3
10 252 28 91 0.3907 6
20 268 12 91 0.5767 11
50 278 2 114 0.1032 23
Bet-and-Run
5 228 52 65 0.0800 3
10 252 28 79 0.3328 5
20 268 12 90 0.4721 9
50 278 2 105 0.0390 18
Hyper
5 228 52 75 0.0781 3
10 252 28 87 0.3309 5
20 268 12 104 0.4710 9
50 278 2 119 0.0385 19
Interestingly, our results dier from [13], where Luby-based restarts per-
formed not as well as Restarts, whereas in our study Bet-and-Run is out-
performed by LubyStat on TSP. This might be due to a dierent approach of
setting tinit and because we use a larger instance set for TSP. Independent of
this Hyper outperforms both.
Figure 1 adds to the results by showing the results of performing single-sided

Wilcoxon rank-sum tests on the outcomes of 10 independent runs. For the two dif-
ferent problem domains, we observe the following. For TSP, Hyper dominates
the eld and is beaten ve times by Luby (as assessed by the statistical tests)
in terms of quality gap to the optimum. For MVC, Hyper typically outperforms
Single, Restarts, and Luby. In contrast to this, Hyper and Bet-and-Run per-
form essentially comparably on MVC, and the dierences are rarely signicant.
Lastly, we summarize the investigations by testing whether the performance
dierences between Hyper and and the other approaches across all instances and
time budgets are statistically signicant. Again, we apply a single-sided Wilcoxon
test to test the null hypothesis that two given distributions are identical. We
compare the approaches based on the performance gap achieved, and across
all time budgets and instances. In particular, we take the median of the 10
independent runs per instance, and then collect for each restart approach the
medians across all instances and time limits. As a consequence, each approach
has 38 6 = 228 medians for TSP and 28 4 = 112 medians for MVC.
Table 3 shows the results of these two tests. In summary, we can deduce
from the outcome that Hyper performs no worse than existing approaches, and
typically better. A closer inspection of the raw results of Hyper and Bet-and-
Run on MVC reveals that their performance is near-identical, despite the fact
that the averages of Hyper are consistently lower than those of Bet-and-Run
(as seen in Table 2). In stark contrast to this, the performance comparisons on
TSP are mostly highly signicant and in an favor of Hyper.
Table 3. One-sided Wilcoxon rank-sum test to test whether the quality gap distrib-
ution of Hyper is shifted to the left of that of the other approaches. Shown are the
p-values.
Single Restarts Luby Bet-and-Run

TSP 0.000003 0.000015 0.478300 0.000002
MVC 0.060172 0.248689 0.354935 0.236808
6 Conclusion
We introduced the idea of learning reactive restart strategies for combinato-
rial search algorithms. We compared this new approach (Hyper) with other
approaches, among them a very recent Bet-and-Run approach that had been
assessed comprehensively on TSP and MVC instances. Across both domains,
Hyper resulted in markedly better average solution qualities, and it exhibited
signicantly increased rates of hitting the best possible solution.
As the investigated problem domains are structurally very dierent, we
expect our approach to generalize to other problem domains as well, such as
continuous and multi-objective optimization problems.
Future work will focus on the development of other runtime features as a
basis for making restart decisions.
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Ecient Adaptive Implementation of the Serial
Schedule Generation Scheme Using
Preprocessing and Bloom Filters
Daniel Karapetyan1(B) and Alexei Vernitski2

1
Institute for Analytics and Data Science, University of Essex, Essex, UK
daniel.karapetyan@gmail.com
2
Department of Mathematical Sciences, University of Essex, Essex, UK
asvern@essex.ac.uk
Abstract. The majority of scheduling metaheuristics use indirect rep-

resentation of solutions as a way to eciently explore the search space.
Thus, a crucial part of such metaheuristics is a schedule generation
scheme procedure translating the indirect solution representation into
a schedule. Schedule generation scheme is used every time a new candi-
date solution needs to be evaluated. Being relatively slow, it eats up most
of the running time of the metaheuristic and, thus, its speed plays signif-
icant role in performance of the metaheuristic. Despite its importance,
little attention has been paid in the literature to ecient implementa-
tion of schedule generation schemes. We give detailed description of serial
schedule generation scheme, including new improvements, and propose
a new approach for speeding it up, by using Bloom lters. The results
are further strengthened by automated control of parameters. Finally, we
employ online algorithm selection to dynamically choose which of the two
implementations to use. This hybrid approach signicantly outperforms
conventional implementation on a wide range of instances.
Keywords: Resource-constrained project scheduling problem Serial

schedule generation scheme Bloom lters Online algorithm selection
1 Introduction
Resource Constrained Project Scheduling Problem (RCPSP) is to schedule a set

of jobs J subject to precedence relationships and resource constraints. RCPSP
is a powerful model generalising several classic scheduling problems such as job
shop scheduling, ow shop scheduling and parallel machine scheduling.
In RCPSP, we are given a set of resources R and their capacities cr , r R.
In each time slot, cr units of resource r are available and can be shared between
jobs. Each job j J has a prescribed consumption vj,r of each resource r R.
We are also given the duration dj of a job j J. A job consumes vj,r units
of resource r in every time slot that it occupies. Once started, a job cannot be
https://doi.org/10.1007/978-3-319-69404-7_9
Ecient Adaptive Implementation of the Serial Schedule Generation Scheme 125
interrupted (no preemption is allowed). Finally, each resource is assigned a set

pred j J of jobs that need to be completed before j can start.
There exist multiple extensions of RCPSP. In the multi-mode extension, each
job can be executed in one of several modes, and then resource consumption and
duration depend on the selected mode. In some applications, resource availability
may vary with time. There could be set-up times associated with certain jobs. In
multi-project extension, several projects run in parallel sharing some but not all
resources. In this paper we focus on the basic version of RCPSP, however some
of our results can be easily generalised to many of its extensions and variations.
Most of the real-world scheduling problems, including RCPSP, are NP-hard,
and hence only problems of small size can be solved to optimality, whereas
for larger problems (meta)heuristics are commonly used. Metaheuristics usually
search in the space of feasible solutions; with a highly constrained problem such
as RCPSP, browsing the space of feasible solutions is hard. Indeed, if a schedule
is represented as a vector of job start times, then changing the start time of a
single job is likely to cause constraint violations. Usual approach is to use indirect
solution representation that could be conveniently handled by the metaheuristic
but could also be eciently translated into the direct representation.
Two translation procedures widely used in scheduling are serial schedule
generation scheme (SSGS) and parallel schedule generation scheme [9]. Some
studies conclude that SSGS gives better performance [7], while others suggest
to employ both procedures within a metaheuristic [10]. Our research focuses on
SSGS.
With SSGS, the indirect solution representation is a permutation of jobs.
The metaheuristic handles candidate solutions in indirect (permutation) form.
Every time a candidate solution needs to be evaluated, SSGS is executed to
translate the solution into a schedule (tj , j J), and only then the objective
value can be computed, see Fig. 1.
Metaheuristic
Candidate solution (permutation )
Objective value SSGS
Candidate schedule (job start times tj , j J)
Objective function
Fig. 1. Classic architecture of a scheduling metaheuristic. To obtain objective value of

a candidate solution, metaheuristic uses SSGS to translate the candidate solution into
a candidate schedule, which is then used by objective function.
SSGS is a simple procedure that iterates through J in the order given by

, and schedules one job at a time, choosing the earliest feasible slot for each
126 D. Karapetyan and A. Vernitski
Algorithm 1. Serial Schedule Generation Scheme (SSGS). Here T is the

upper bound on the makespan.
input : Solution in permutation form, respecting precedence relations
output : Schedule tj , j J
1 At,r cr for every t = 1, 2, . . . , T and r R;
2 for i = 1, 2, . . . , |J| do
3 j (i);
4 t0 maxj pred j tj + dj ;
5 tj find (j, t0 , A) (see Algorithm 2);
6 update(j, tj , A) (see Algorithm 3);
7 return tj , j J;
Algorithm 2. Conventional implementation of find (j, t0 , A) a function

to nd the earliest feasible slot for job j.
input : Job j J to be scheduled
input : Earliest start time t0 as per precedence relations
input : Current availability A of resources
output : Earliest feasible start time for job j
1 tj t0 ;
2 t tj ;
3 while t < tj + dj do
4 if At,r vj,r for every r R then
5 t t + 1;
6 else
7 tj t + 1;
8 t tj ;
9 return tj ;
job. The only requirement for is to respect the precedence relations; otherwise
SSGS produces a feasible schedule for any permutation of jobs. The pseudo-
code of SSGS is given in Algorithm 1, and its two subroutines find and update
in Algorithms 2 and 3.
Commonly, the objective of RCPSP is to nd a schedule that minimises
the makespan, i.e. the time required to complete all jobs; however other objec-
tive functions are also considered in the literature. We say that an objective
function of a scheduling problem is regular if advancing the start time of a
job cannot worsen the solutions objective value. Typical objective functions of
RCPSP, including makespan, are regular. If the scheduling problem has a reg-
ular objective function, then SSGS guarantees to produce active solutions, i.e.
solutions that cannot be improved by changing tj for a single j J. Moreover,
it was shown [8] that for any active schedule S there exists a permutation
for which SSGS will generate S. Since any optimal solution S is active, search-
ing in the space of feasible permutations is sucient to solve the problem.
Algorithm 3. Procedure update(j, tj , A) to update resource availability A

after scheduling job j at time tj .
input : Job j and its start time tj
input : Resource availability A
1 for t tj , tj + 1, . . . , tj + dj 1 do
2 Aj,r Aj,r vj,r for every r R;
This is an important property of SSGS; the parallel schedule generation scheme,

mentioned above, does not provide this guarantee [8] and, hence, may not in some
circumstances allow a metaheuristic nding optimal or near-optimal solutions.
The runtime of a metaheuristic is divided between its search control mecha-
nism that modies solutions and makes decisions such as accepting or rejecting
new solutions, and SSGS. While SSGS is a polynomial algorithm, in practice it
eats up the majority of the metaheuristic runtime (over 98% as reported in [1]).
In other words, by improving the speed of SSGS twofold, one will (almost) double
the number of iterations a metaheuristic performs within the same time budget,
and this increase in the number of iterations is likely to have a major eect on
the quality of obtained solutions.
In our opinion, not enough attention was paid to SSGS a crucial component
of many scheduling algorithms, and this study is to close this gap. In this paper
we discuss approaches to speed up the conventional implementation of SSGS.
Main contributions of our paper are:
A detailed description of SSGS including old and new speed-ups (Sect. 2).
New implementation of SSGS employing Bloom lters for quick testing of
resource availability (Sect. 3).
A hybrid control mechanism that employs intelligent learning to dynamically
select the best performing SSGS implementation (Sect. 4).
Empirical evaluation in Sect. 5 conrms that both of our implementations of
SSGS perform signicantly better than the conventional SSGS, and the hybrid
control mechanism is capable of correctly choosing the best implementation while
generating only negligible overheads.
2 SSGS Implementation Details

Before we proceed to introducing our main new contributions in Sects. 3 and 4,
we describe what state-of-the-art implementation of SSGS we use, including
some previously unpublished improvements.
2.1 Initialisation of A
The initialisation of A in line 1 of Algorithm 1 iterates through T slots, where T is
the upper bound on the makespan. It was noted in [1] that instead of initialising
Algorithm 4. Enhanced implementation of find (j, t0 , A)

input : Job j J to be scheduled
input : Earliest start time t0 as per precedence relations
input : Current availability A of resources
output : Earliest feasible start time for job j
1 tj t0 ;
2 t tj + dj 1;
3 ttest tj ;
4 while t ttest do
5 if At,r Vj,r for every r R then
6 t t 1;
7 else
8 ttest tj + dj ;
9 tj t + 1;
10 t tj + dj 1;
11 return tj ;
A at every execution of SSGS, one can reuse this data structure between the
executions. To correctly initialise A, at the end of SSGS we restore At,r for each
r R and each slot where some job was scheduled: At,r cr for r R and
t = 1, 2, . . . , M , where M is the makespan of the solution. Since M T and
usually M T , this notably improves the performance of SSGS [1].
2.2 Ecient Search of the Earliest Feasible Slot for a Job
The function find (j, t0 , I, A) nds the earliest slot feasible for scheduling job
j. Its conventional implementation (Algorithm 2) takes O(T |R|) time, where T
is the upper bound of the time horizon. Our enhanced implementation of find
(Algorithm 4), rst proposed in [1], has the same worst case complexity but is
more ecient in practice. It is inspired by the Knuth-Morris-Pratt substring
search algorithm. Let tj be the assumed starting time of job J. To verify if it is
feasible, we need to test suciency of resources in slots tj , tj + 1, . . . , tj + dj 1.
Unlike the conventional implementation, our enhanced version tests these slots
in the reversed order. The order makes no dierence if the slot is feasible, but
otherwise testing in reversed order allows us to skip some slots; in particular, if
slot t is found to have insucient resources then we know that feasible tj is at
least t + 1.
A further speed up, which was not discussed in the literature before, is to
avoid re-testing of slots with sucient resources. Consider the point when we
nd that the resources in slot t are insucient. By that time we know that the
resources in t + 1, t + 2, . . . , tj + dj 1 are sucient. Our heuristic is to remember
that the earliest slot ttest to be tested in future iterations is tj + dj .
2.3 Preprocessing and Automated Parameter Control

We observe that in many applications, jobs are likely to require only a subset
of resources. For example, in construction works, to dig a hole one does not
need cranes or electricians, hence the dig a hole job will not consume those
resources. To exploit this observation, we pre-compute vector Rj of resources
used by job j, and then iterate only through resources in Rj when testing resource
suciency in find (Algorithm 4, line 5) and updating resource availability in
update (Algorithm 3, line 2). Despite the simplicity of this idea, we are not aware
of anyone using or mentioning it before.
We further observe that some jobs may consume only one resource. By cre-
ating specialised implementations of find and update, we can reduce the depth
of nested loops. While this makes no dierence from the theoretical point of
view, in practice this leads to considerable improvement of performance. Cor-
rect implementations of find and update are identied during preprocessing and
do not cause overheads during executions of SSGS.
Having individual vectors Rj of consumed resources for each job, we can also
intelligently learn the order in which resource availability is tested (Algorithm 4,
line 5). By doing this, we are aiming at minimising the expected number of iter-
ations within the resource availability test. For example, if resource r is scarce
and job j requires signicant amount of r, then we are likely to place r at the
beginning of Rj . More formally, we sort Rj in descending order of probability
that the resource is found to be insucient during the search. This probabil-
ity is obtained empirically by a special implementation of SSGS which we call
SSGSdata . SSGSdata is used once to count how many times each resource turned
out to be insucient during scheduling of a job. (To avoid bias, SSGSdata tests
every resource in Rj even if the test could be terminated early.) After a single
execution of SSGSdata , vectors Rj are optimised, and in further executions default
implementation of SSGS is used.
One may notice that the data collected in the rst execution of SSGS may
get outdated after some time; this problem in addressed in Sect. 4.
Ordering of Rj is likely to be particularly eective on instances with asym-
metric use of resources, i.e. on real instances. Nevertheless, we observed improve-
ment of runtime even on pseudo-random instances as reported in Sect. 5.
3 SSGS Implementation Using Bloom Filters

Performance bottleneck of an algorithm is usually its innermost loop. Observe
that the innermost loop of the find function is the test of resource suciency in
a slot, see Algorithm 4, line 5. In this section we try to reduce average runtime
of this test from O(|R|) to O(1) time. For this, we propose a novel way of using
a data structure known as Bloom lter.
Bloom lters were introduced in [2] as a way of optimising dictionary lookups,
and found many applications in computer science and electronic system engineer-
ing [3,13]. Bloom lters usually utilise pseudo-random hash functions to encode
data, but in some applications [6] non-hash-based approaches are used. In our
paper, we also use a non-hash-based approach, and to our knowledge, our paper
is the rst in which the structure of Bloom lters is chosen dynamically accord-
ing to the statistical properties of the data, with the purpose of improving the
speed of an optimisation algorithm.
In general, Bloom lters can be dened as a way of using data, and they
are characterised by two aspects: rst, all data is represented by short binary
arrays of a xed length (perhaps with a loss of accuracy); second, the process of
querying data involves only bitwise comparison of binary arrays (which makes
querying data very fast).
We represent both each jobs resource consumption and resource availability
at each time slot, by binary arrays of a xed length; we call these binary arrays
Bloom lters. Our Bloom lters will consist of bits which we call resource level
bits. Each resource bit, denoted by ur,k , r R, k {1, 2, . . . , cr }, means k units
of resource r (see details below). Let U be the set of all possible resource bits.
A Bloom filter structure is an ordered subset L U , see Fig. 2 for an example.
Suppose that a certain Bloom lter structure L is xed. Then we can introduce
B L (j), the Bloom lter of job j, and B L (t), the Bloom lter of time slot t, for
each j and t. Each B L (j) and B L (t) consists of |L| bits dened as follows: if ur,k
is the ith element of L then

1 if v j,r k, 1 if At,r k,
B L (j)i = and B L (t)i =
0 otherwise, 0 otherwise.
u1,2 u1,3 u1,4 u2,1 u2,3 u3,1 u3,3 u3,4

2 3 4 1 3 1 3 4
Resource 1 Resource 2 Resource 3
Fig. 2. Example of a Bloom lter structure for a problem with 3 resources, each having
capacity 4.
To query if a job j can be scheduled in a time slot t, we compare Bloom

lters B L (j) and B L (t) bitwise; then one of three situations is possible, as the
following examples show. Consider a job j and three slots, t, t and t , with the
following resource consumption/availabilities, and Bloom lter structure as in
Fig. 2:
vj,1 = 3 vj,2 = 2 vj,3 = 0 B L (j) = (110 10 000)
At,1 = 2 At,2 = 3 At,3 = 4 B L (t) = (100 11 111)
At ,1 = 3 At ,2 = 1 At ,3 = 4 B L (t ) = (110 10 111)
At ,1 = 3 At ,2 = 2 At ,3 = 2 B L (t ) = (110 10 100)
For two bit arrays of the same length, let notation mean bitwise less or
equal. By observing that B L (j) B L (t), we conclude that resources in slot t
are insucient for j; this conclusion is guaranteed to be correct. By observing

that B L (j) B L (t ), we conclude tentatively that resources in slot t may be
sucient for j; however, further verication of the complete data related to j
and t (that is, vj, and At , ) is required to get a precise answer; one can see
that vj,2 At ,2 , hence this is what is called a false positive. Finally, we observe
that B L (j) B L (t ), and a further test (comparing vj, and At , ) conrms that
resources in t are indeed sucient for j.
Values of B L (j), j J, are pre-computed when L is constructed, and B L (t),
t = 1, 2, . . . , T , are maintained by the algorithm.
3.1 Optimisation of Bloom Filter Structure
The length |L| of a Bloom lter is limited to reduce space requirements and,
more importantly for our application, speed up Bloom lter tests. Note that if
|L| is small (such as 32 or 64 bits) then we can exploit ecient bitwise operators
implemented by all modern CPUs; then each Bloom lter test takes only one
CPU operation. We set |L| = 32 in our implementation. While obeying this
constraint, we aim at minimising the number of false positives, because false
positives slow down the implementation.
u1,1 u1,2 u1,3 u1,4 u2,1 u2,2 u2,3 u2,4 u3,1 u3,2 u3,3 u3,4
1 2 3 4 1 2 3 4 1 2 3 4
Resource 1 Resource 2 Resource 3
Fig. 3. Example of a Bloom lter structure for a problem with 3 resources, each having
capacity 4, with L = U .
Our L building algorithm is as follows:
1. Start with L = U , such as in Fig. 3.

2. If |L| is within the prescribed limit, stop.
3. Otherwise select ur,k L that is least important and delete it. Go to step 2.
By least important we mean that the deletion of it is expected to

have minimal impact of the expected number of false positives. Let L =
(. . . , ur,q , ur,k , ur,m , . . .). Consider a job j such that k vj,r < m and a slot
t such that q At,r < k. With L as dened above, Bloom lters correctly iden-
tify that resources in slot t are insucient for job j: B L (j) B L (t). However,
without the resource level bit ur,k we get a false positive: B L (j) B L (t). Thus,
the probability of false positives caused by deleting ur,k from L in is as follows:
m1 i1

Dkr Ekr ,
k=i k=q
where Dkr is the probability that a randomly chosen job needs exactly k units
of resource r, and Ekr is the probability that a certain slot, when we examine
it for scheduling a job, has exactly k units of resource r available. The proba-
bility distribution Dr is produced from the RCPSP instance data during pre-
processing.1 The probability distribution E r is obtained empirically during the
run of SSGSdata (see Sect. 2.2); each time resource suciency is tested within
SSGSdata , its availability is recorded.
3.2 Additional Speed-ups
While positive result of a Bloom lter test generally requires further verication
using full data, in some circumstances its correctness can be guaranteed. In
particular, if for some r R and j J we have ur,k L and vj,r = k, then the
Bloom lter result, whether positive or negative, does not require verication.
Another observation is that updating B L (t) in update can be done in O(|Rj |)
operations instead of O(|R|) operations. Indeed, instead of computing B L (t)
from scratch, we can exploit our structure of Bloom lters. We update each bit
related to resources r Rj , but we keep intact other bits. With some trivial
pre-processing, this requires only O(|Rj |) CPU operations.
We also note that if |Rj | = 1, i.e. job j uses only one resource, then Bloom
lters will not speed up the find function for that job and, hence, in such cases
we use the standard find function specialised for one resource (see Sect. 2.2).
4 Hybrid Control Mechanism

So far we have proposed two improved implementations of SSGS: one using
Bloom lters (which we denote SSGSBF ), and the other one not using Bloom
lters (which we denote SSGSNBF ). While it may look like SSGSBF should always
be superior to SSGSNBF , in practice SSGSNBF is often faster. Indeed, Bloom lters
usually speed up the find function, but they also slow down update, as in SSGSBF
we need to update not only the values At,r but also the Bloom lters B L (t)
encoding resource availability. If, for example, the RCPSP instance has tight
precedence relations and loose resource constraints then find may take only a
few iterations, and then the gain in the speed of find may be less than the loss
of speed of update. In such cases SSGSBF is likely to be slower then SSGSNBF .
In short, either of the two SSGS implementations can be superior in certain
circumstances, and which one is faster mostly depends on the problem instance.
In this section we discuss how to adaptively select the best SSGS implementation.
Automated algorithm selection is commonly used in areas such as sorting, where
multiple algorithms exist. A typical approach is then to extract easy to compute
input data features and then apply o-line learning to develop a predictor of
which algorithm is likely to perform best, see e.g. [5]. With sorting, this seems
1
In multi-mode extension of RCPSP, this distribution depends on selected modes and
hence needs to be obtained empirically, similarly to how we obtain E r .
to be the most appropriate approach as the input data may vary signicantly
between executions of the algorithm. Our case is dierent in that the most
crucial input data (the RCPSP instance) does not change between executions
of SSGS. Thus, during the rst few executions of SSGS, we can test how each
implementation performs, and then select the faster one. This is a simple yet
eective control mechanism which we call Hybrid.
Hybrid is entirely transparent for the metaheuristic; the metaheuristic simply
calls SSGS whenever it needs to evaluate a candidate solution and/or generate
a schedule. The Hybrid control mechanism is then intelligently deciding each
time which implementation of SSGS to use based on information learnt during
previous runs.
An example of how Hybrid performs is illustrated in Fig. 4. In the rst exe-
cution, it uses SSGSdata to collect data required for both SSGSBF and SSGSNBF . For
the next few executions, it alternates between SSGSBF and SSGSNBF , measuring
the time each of them takes. During this stage, Hybrid counts how many times
SSGSBF was faster than the next execution of SSGSNBF . Then we use the sign
test [4] to compare the implementations. If the dierence is signicant (we use
a 5% signicance level for the sign test) then we stop alternating the implemen-
tations and in future use only the faster one. Otherwise we continue alternating
the implementations, but for at most 100 executions. (Without such a limitation,
there is a danger that the alternation will never stop if the implementations
perform similarly; since there are overheads associated with the alternation and
time measurement, it is better to pick one of the implementations and use it in
future executions.)
Restart
First 10,000 executions Next 10,000 executions
data BF NBF BF NBF BF BF BF BF data BF NBF BF NBF
Choosing SSGS impl. Running the faster impl.
Special execution to collect data to optimise Rj and Bloom lters
Fig. 4. Stages of the Hybrid control mechanism. Each square shows one execution of
SSGS, and the text inside describes which implementation of SSGS is used. SSGSdata
is always used in the rst execution. Further few executions (at most 100) alternate
between SSGSBF and SSGSNBF , with each execution being timed. Once sign test shows
signicant dierence between the SSGSBF and SSGSNBF implementations, the faster one
is used for the rest of executions. After 10,000 executions, previously collected data is
erased and adaptation starts from scratch.
Our decision to use the sign test is based on two considerations: rst, it is very
fast, and second, it works for distributions which are not normal. This makes
our approach dierent from [12] where the distributions of runtimes are assumed
to be normal. (Note that in our experiments we observed that the distribution
of running times of an SSGS implementation resembles Poisson distribution.)
As pointed out in this and previous sections, optimal choices of parameters of
the SSGS implementations mostly depend on the RCPSP instance which does
not change throughout the metaheuristic run; however solution also aects the
performance. It should be noted though that metaheuristics usually apply only
small changes to the solution at each iteration, and hence solution properties tend
to change relatively slowly over time. Consequently, we assume that parameters
chosen in one execution of SSGS are likely to remain ecient for some further
executions. Thus, Hybrid restarts every 10,000 executions, by which we mean
that all the internal data collected by SSGS is erased, and learning starts from
scratch, see Fig. 4. This periodicity of restarts is a compromise between accuracy
of choices and overheads, and it was shown to be practical in our experiments.
5 Empirical Evaluation
In this Section we evaluate the implementations of SSGS discussed above. To

replicate conditions within a metaheuristic, we designed a simplied version of
Simulated Annealing. In each iteration of our metaheuristic, current solution
is modied by moving a randomly selected job into a new randomly selected
position (within the feasible range). If the new solution is not worse than the
previous one, it is accepted. Otherwise the new solution is accepted with 50%
probability. Our metaheuristic performs 1,000,000 iterations before terminating.
We evaluate SSGSBF , SSGSNBF and Hybrid. These implementations are com-
pared to conventional SSGS, denoted by SSGSconv , which does not employ any
preprocessing or Bloom lters and uses conventional implementation of find
(Algorithm 2).
We found that instances in the standard RCPSP benchmark set PSPLIB [11]
occupy a relatively small area of the feature space. For example, all the RCPSP
instances in PSPLIB have exactly four resources, and the maximum job duration
is always set to 10. Thus, we chose to use the PSPLIB instance generator, but
with a wider range of settings. Note that for this study, we modied the PSPLIB
instance generator by allowing jobs not to have any precedence relations. This
was necessary to extend the range of network complexity parameter (to include
instances with scarce precedence relations), and to speed up the generator, as the
original implementation would not allow us to generate large instances within
reasonable time.
All the experiments are conducted on a Windows Server 2012 machine based
on Intel Xeon E5-2690 v4 2.60 GHz CPU. No concurrency is employed in any of
the implementations or tests.
To see the eect of various instance features on SSGS performance, we select
one feature at a time and plot average SSGS performance against the values
of that feature. The rest of the features (or generator parameters) are then set
as follows: number of jobs 120, number of resources 4, maximum duration of
job 10, network complexity 1, resource factor 0.75, and resource strength 0.1.
These values correspond to some typical settings used in PSPLIB. For formal
denitions of the parameters we refer to [11].
For each combination of the instance generator settings, we produce 50
instances using dierent random generator seed values, and in each of our exper-
iments the metaheuristic solves each instance once. Then the runtime of SSGS is
said to be the overall time spent on solving those 50 instances, over 50,000,000
(which is the number of SSGS executions). The metaheuristic overheads are
relatively small and are ignored.
From the results reported in Fig. 5 one can see that our implementations of
SSGS are generally signicantly faster than SSGSconv , but performance of each
implementation varies with the instance features. In some regions of the instance
space SSGSBF outperforms SSGSNBF , whereas in other regions SSGSNBF outperforms
SSGSBF . The dierence in running times is signicant, up to a factor of two in our
experiments. At the same time, Hybrid is always close to the best of SSGSBF and
SSGSNBF , which shows eciency of our algorithm selection approach. In fact, when
SSGSBF and SSGSNBF perform similarly, Hybrid sometimes outperforms both; this
behaviour is discussed below.
Another observation is that SSGSNBF is always faster than SSGSconv (always
below the 100% mark) which is not surprising; indeed, SSGSNBF improves the per-
formance of both find and update. In contrast, SSGSBF is sometimes slower than
SSGSconv ; on some instances, the speed-up of the find function is overweighed
by overheads in both find and update. Most important though is that Hybrid
outperforms SSGSconv in each of our experiments by 8 to 68%, averaging at 43%.
In other words, within a xed time budget, an RCPSP metaheuristic employ-
ing Hybrid will be able to run around 1.8 times more iterations than if it used
SSGSconv .
To verify that Hybrid exhibits the adaptive behaviour and does not just stick
to whichever implementation has been chosen initially, we recorded the imple-
mentation it used in every execution for several problems, see Fig. 6. For this
experiment, we produced three instances: rst instance has standard parameters
except Resource Strength is 0.2; second instance has standard parameters except
Resource Factor is 0.45; third instance has standard parameters except Maxi-
mum Job Duration is 20. These parameter values were selected such that the two
SSGS implementations would be competitive and, therefore, switching between
them would be a reasonable strategy. One can see that the switches occur several
times throughout the run of the metaheuristic, indicating that Hybrid adapts to
the changes of solution. For comparison, we disabled the adaptiveness and mea-
sured the performance if only implementation chosen initially is used throughout
all iterations; the results are shown on Fig. 6. We conclude that Hybrid benets
from its adaptiveness.
SSGSBF SSGSNBF Hybrid
65
70
Runtime, %
60
60
55
50
50
0 200 400 600 800 1,000 2 4 6 8 10
Number of jobs Number of resources
150
70
Runtime, %
60
100
50
50
40
0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Resource strength Resource factor
70
100
Runtime, %
60 80
60
50
0 0.5 1 1.5 2 2.5 0 10 20 30

Network complexity Maximum job duration
Fig. 5. These plots show how performance of the SSGS implementations depends on
various instance features. Vertical axis gives the runtime of each implementation rela-
tive to SSGSconv . (SSGSconv graph would be a horizontal line y = 100%.)
SSGSBF SSGSNBF
99%
96%
95%
executions
0 1 000 000
Fig. 6. This diagram shows how Hybrid switches between SSGS implementations while
solving three problem instances. The number on the right shows the time spent by
Hybrid compared with the time that would be needed if only the implementation
chosen at the start would be used for all iterations.
6 Conclusions and Future Work
In this paper we discussed the crucial component of many scheduling metaheuris-

tics, the serial schedule generation scheme (SSGS). SSGS eats up most of the
runtime in a typical scheduling metaheuristic, therefore performance of SSGS is
critical to the performance of the entire metaheuristic, and thus each speed-up
of SSGS has signicant impact. We described existing and some new speed-ups
to SSGS, including preprocessing and automated parameter control. This imple-
mentation clearly outperformed the conventional SSGS in our experiments.
We further proposed a new implementation that uses Bloom lters, particularly
ecient in certain regions of the instance space. To exploit strengths of both
implementations, we proposed a hybrid control mechanism that learns the per-
formance of each implementation and then adaptively chooses the SSGS version
that is best for a particular problem instance and phase of the search. Exper-
iments showed that this online algorithm selection mechanism is eective and
makes Hybrid our clear choice. Note that Hybrid is entirely transparent for the
metaheuristic which uses it as if it would be simple SSGS; all the learning and
adaptation is hidden inside the implementation.
The idea behind online algorithm selection used in this project can be further
developed by making the number of executions between restarts adaptable. To
determine the point when the established relation between the SSGS implemen-
tations may have got outdated, we could treat the dynamics of the implementa-
tions performance change as two random walks, and use the properties of these
two random walks to predict when they may intersect.
All the implementations discussed in the paper, in C++, are available
for downloading from http://csee.essex.ac.uk/sta/dkarap/rcpsp-ssgs.zip. The
implementations are transparent and straightforward to use.
While we have only discussed SSGS for the simple RCPSP, our ideas can
easily be applied in RCPSP extensions. We expect some of these ideas to work
particularly well in multi-project RCPSP, where the overall number of resources
is typically large but only a few of them are used by each job.
Acknowledgements. We would like to thank Prof. Rainer Kolisch for providing

us with a C++ implementation of the PSPLIB generator. It should be noted that,
although the C++ implementation was developed to reproduce the original Pascal
implementation, the exact equivalence cannot be guaranteed; also, in our experiments
we used a modication of the provided C++ code.
References
1. Asta, S., Karapetyan, D., Kheiri, A., Ozcan, E., Parkes, A.J.: Combining Monte-
Carlo and hyper-heuristic methods for the multi-mode resource-constrained multi-
project scheduling problem. Inf. Sci. 373, 476498 (2016)
2. Bloom, B.H.: Space/time trade-os in hash coding with allowable errors. Commun.
ACM 13(7), 422426 (1970)
3. Broder, A., Mitzenmacher, M.: Network applications of bloom lters: a survey.
Internet Math. 1(4), 485509 (2004)
4. Cohen, L., Holliday, M.: Practical Statistics for Students: An Introductory Text.
Paul Chapman Publishing Ltd., London (1996)
5. Guo, H.: Algorithm selection for sorting and probabilistic inference: a machine
learning-based approach. Ph.D. thesis, Kansas State University (2003)
6. Kayaturan, G.C., Vernitski, A.: A way of eliminating errors when using Bloom
lters for routing in computer networks. In: Fifteenth International Conference on
Networks, ICN 2016, pp. 5257 (2016)
7. Kim, J.-L., Ellis Jr., R.D.: Comparing schedule generation schemes in resource-
constrained project scheduling using elitist genetic algorithm. J. Constr. Eng.
Manag. 136(2), 160169 (2010)
8. Kolisch, R.: Serial and parallel resource-constrained project scheduling methods
revisited: theory and computation. Eur. J. Oper. Res. 90(2), 320333 (1996)
9. Kolisch, R., Hartmann, S.: Heuristic algorithms for the resource-constrained
project scheduling problem: classication and computational analysis. In: Weglarz,

J. (ed.) Project Scheduling, pp. 147178. Springer, Boston (1999). doi:10.1007/
978-1-4615-5533-9 7
10. Kolisch, R., Hartmann, S.: Experimental investigation of heuristics for resource-
constrained project scheduling: an update. Eur. J. Oper. Res. 174(1), 2337 (2006)
11. Kolisch, R., Sprecher, A.: PSPLIB - a project scheduling problem library: OR
software - ORSEP operations research software exchange program. Eur. J. Oper.
Res. 96(1), 205216 (1997)
12. Lau, J., Arnold, M., Hind, M., Calder, B.: Online performance auditing: using hot
optimizations without getting burned. SIGPLAN Not. 41(6), 239251 (2006)
13. Tarkoma, S., Rothenberg, C.E., Lagerspetz, E.: Theory and practice of bloom lters
for distributed systems. IEEE Commun. Surv. Tutor. 14(1), 131155 (2012)
Interior Point and Newton Methods in Solving
High Dimensional Flow Distribution Problems
for Pipe Networks
Oleg O. Khamisov1(B) and Valery A. Stennikov2

1
Skolkovo Institute of Science and Technology, Moscow, Russia
oleg.khamisov@skolkovotech.ru
2
Melentiev Energy Systems Institute SB RAS, Irkutsk, Russia
sva@isem.irk.ru
Abstract. In this paper optimal ow distribution problem in pipe

network is considered. The investigated problem is a convex sparse
optimization problem with linear equality and inequality constrains.
Newton method is used for problem with equality constrains only and
obtains an approximate solution, which may not satisfy inequality con-
straints. Then Dikin Interior Point Method starts from the approximate
solution and nds an optimal one. For problems of high dimension sparse
matrix methods, namely Conjugate Gradient and Cholesky method with
nested dissection, are applied. Since Dikin Interior Point Method works
much slower then Newton Method on the matrices of big size, such app-
roach allows us to obtain good starting point for this method by using
comparatively fast Newton Method. Results of numerical experiments
are presented.
Keywords: Pipe network Convex optimization Newton Method

Interior Point Method Sparse matrix Large-scale optimization
1 Introduction
In this paper optimal (steady-state) ow distribution problem in pipe network is
considered. From mathematical point of view this problem is sparse large-scale
convex optimization problem.
Several approaches to nd steady-state solution exist. A comprehensive sur-
vey of methods is presented in [6]. Our paper uses approach similar to [2], but
also includes sparse matrices techniques in order to eciently solve high dimen-
sional ow distribution problems.
The aim of this paper is application of new technology for solving the con-
sidered problem.
2 Problem Statement
Let us consider pipe network, which is given by its incidence matrix
A IR(n + 1)m , where m is the number of oriented edges and n+1 is the number
https://doi.org/10.1007/978-3-319-69404-7_10
140 O.O. Khamisov and V.A. Stennikov
n
of nodes. Vector of nodal ow rates Q IRn+1 , i = 1 Qi = 0, with components
Qi > 0 for sources, Qi < 0 for sinks and Qi = 0 for the nodes of connection.
The ow distribution problem has the following statement [5]:
F (x) min, x IRm , (1)
Ax = Q, (2)
here x is the vector of unknown ows on the edges, that have to be found: xi > 0
if ow coincides with the direction of the edge i, otherwise xi < 0. Function F
is a twice dierentiable convex function, which will be described below.
System (2) describes the rst Kirchhos law. Since A is an incidence matrix,
we have rank A = n, therefore it is possible to exclude one arbitrary row and
get equivalent system:
Ax = Q, (3)
here A IRnm , and Q IRn are new matrix and right hand side vector,
obtained by such exclusion. This is done to avoid working with singular matrix.
The objective function is dened in the following way [2,5]:
m
Si |xi |i
F (x) = Hi xi , (4)
i=1
i
here Si are edge resistance coecients, i 3 are integer numbers, Hi are

pressure coecients. Usually all i are assumed to be equal each other, in [2]
all i = = 3. Further calculations are done under this assumption, therefore
objective function can be presented in the form
m
Si |xi |3
F (x) = Hi xi .
i=1
3
For the problem (1) with linear constraints (3) Newton Method can be
applied with certain conditions, which are considered below.
If line contains a pump, xi must be nonnegative. Let I1 {1, . . . , m} be a
set of lines with pumps without ow limitations. Additionally, ow values on
some lines i I2 {1, . . . , m}, which can also contain pumps, must be limited
from above by i > 0. Note that I1 I2 = . Therefore, the following inequality
constraints on the ows have to be taken into consideration
xi 0, i I1 , i xi i , i I2 , (5)
here, if pump is present on the edge i I2 then i = 0, otherwise i = i . When

(5) is present, the solution of problem (1) and (3) is found by Newton Method and
then Dikin Interior Point method is applied to take into consideration inequality
constraints (5).
Interior Point and Newton Methods in Solving High Dimensional Flow 141
3 Newton Method
Let us consider problem (1) and (3). The Lagrange function for this system has
the following form:
L(x, ) = F (x) + T (Ax Q), IRn .
Due to the convexity of the objective function, equivalence of the Lagrange func-
tion gradient in x to zero is a sucient minimum condition. Therefore Newton
Method is applied to solve the equation
L
= 0.
x
At (k + 1) step of the Newton method vector xk + 1 is calculated according to
the following formula [8]
k+1 k
x x 1
k+1 = k 2 L(xk , k ) L(xk , k ),

where

F (x) + AT 2 F (x) AT
L(x, ) = , L(x, ) =
2
.
Ax Q A 0
Therefore, xk + 1 is obtained from the following system of equations

2 k+1 2
F (xk ) AT x F (xk )xk F (xk )
= , (6)
A 0 k + 1 Q
where 2 F (x) is objective function Hessian.
2 F (x) = 2diag(S1 |x1 |, . . . , Sm |xm |).
During each iteration it is necessary to nd a solution of system of linear

equations with symmetric indenite matrix, therefore its solution can be found
by Conjugate Gradient method (Algorithm 1) [9,10]. Denote by y k IRn+m
vector, consisting of xk and k . The pair xk , k obtained at the previous iteration
of the Newton Method is used as starting approximation, to reduce the number
of iterations in Conjugate Gradient method. Hessian 2 L(xk , k ) is not a posi-
tive denite matrix, moreover it can be singular, therefore the algorithm can fail
if qi = 0 on the line 3 of Algorithm 1. In this case Conjugate Gradient Method
and Newton Methods stop without nding solution and, starting from the cur-
rent approximation, we switch to Interior Point Method which is described in
the next section. Otherwise Conjugate Gradient Method stops, when solution of
linear system is found and Newton Method continues working.
Algorithm 1. Conjugate Gradient

1: p0 = r0 = L(xk , k ) 2 L(xk , k )y k 1
2: for i = 0, 1, . . . , until ri + 1 < do
3: qi = (pi )T 2 L(xk , k )pi
4: if qi = 0 then
5: exit (Start interior point search)
6: end if i T i
7: i = (r q)i r
8: y k,i + 1 = y k,i + i pi
9: ri + 1 = ri i 2 L(xk , k )pi
(r i + 1 )T r i + 1
10: i = (r i )T r i
11: pi + 1 = ri + 1 + i pi
12: end for
Algorithm 2. Interior point search

1: Take initial point x0 such that it strictly satises (5)
2: Take r0 such that r0
3: for k = 1,2 . . . , until rk < do
k 2
(xi ) , i I1 ,
k
4: i = min{(xk i )2 , (xk i )2 }, i I2

1, i I1 I2
5: Dk = diag(1k , . . . , m
k
)
6: B k = ADk AT
7: rk = Axk Q
8: Find uk as the solution of the system B k u = rk
9: k = uk A
10: g k = Dk k
11: k = min{1, k }
12: xk + 1 = xk + k g k
13: end for
4 Interior Point Method

Consider the problem (1), (3) and (5). In order to solve it by Dikin Interior
Point Method [1,3,4] it is necessary to nd interior point of the set given by
constraints (3) and (5). Search for interior point is done by the Algorithm 2. On
the line 11, k is maximal feasible step length with respect to (5), (0, 1) is a
coecient insuring that xk+1 is an interior point. Based on [1] it is taken equal
to 0.8, is the given tolerance.
After interior point was found, Interior Point Method (Algorithm 3) which
is similar to Algorithm 2, starts working. On the line 12, k is found as a result
of exact one-dimensional minimization of convex function () = F (xk + g k ).
Derivative of has the following form:
m

() = gik Si |xki + gik |2 sign(xki + gik ) Hi .
i=1
Algorithm 3. Interior Point Method

1: Take initial point x0 so that it strictly satises constraints (3), (5)
2:
3: Take 0 such that 0
4: for k = 1,2 . . . , until k < do
k 2
(xi ) , i I1 ,
5: ik = min{(xk i )2 , (xk i )2 }, i I2 ,

1, i I1 I2
6: D = diag(1k , . . . , m
k k
)
7: W k = ADk , B k = W k AT
8: ck = F (xk ), dk = W k ck
9: Find uk as the solution of the system B k u = dk
10: k = uk A ck
11: g k = Dk k
12: k = arg min {F (xk + k g k )}
[0,1]
13: k = min{k , k }
14: xk + 1
= xk + k g k
15: k = n k k 2
i = 1 i (i )
16: end for
Let us use the following notation: ik is a root of the equation
Si |xki + gik |2 sign(xki + gik ) Hi = 0, i K = {i : gik = 0}.
As can be seen, ik have the following form:

1

Hi 2 k Hi
Si x i gik , Si 0,
i =
k
12

H Si
i
xk
i gik , Hi
Si < 0.
Function () is monotonously increasing, therefore solution k of the equation

() = 0 belongs to the interval [miniK ik , maxiK ik ]. Search for k on this
interval is done by cubic interpolation method [7].
On the line 13, k is dened in the same way as is in Algorithm 2.
As it can be seen, the most computationally intensive part in the both algo-
rithms is the solution of linear systems. Since B k is always positive denite,
sparse Cholesky method is applicable here. As it is shown in [1113] the sparse
Cholesky method consists of the following steps:
1. Fill-in reduction (permutation of the matrix B aimed to reduce amount of
nonzero elements in Cholesky matrix). In this paper nested dissection [10,11]
is used as the ll-in reduction technique.
2. Symbolic factorization (search for indexes of nonzero elements in Cholesky
matrix).
3. Numerical factorization (calculation of the Cholesky matrix nonzero ele-
ments).
In the both algorithms we have B k = ADk AT , so at each iteration only matrix

Dk is changing. It does not aect the nonzero structure of B k , which is the same
as the structure of AAT , therefore ll-in reduction and symbolic factorization can
be done before the cycle. Additionally numerical factorization can be done not at
every iteration, but only if change of xk becomes small or when change xk results
in constrains violation. Moreover, if ll-in reduction and symbolic factorization
are done during the interior point search, this information can be used in Interior
Point Method, as well as numerical factorization during several rst iterations.
Optimized algorithm, that unies interior point search and Interior Point method
is given by the Algorithm 4. Computational eect of such approach is described
in the section Numerical result.
Algorithm 4. Interior Point Method optimized

1: Take initial point x0 so that it strictly satises (5)
2: Obtain permutation, granted by ll-in reduction, based on the nonzero structure
of AAT
3: Do symbolic factorization.
4: (Start interior point search)
6: Choose g 0 such that g0 > , 0 = 1.
7: for k = 1, 2, . . . until k < do
8: if k1 g k 1 then
9: Calculate Dk
10: Refresh B k and do numerical factorization
11: end if
12: Calculate uk using already obtained Cholesky decomposition
13: Calculate xk + 1 and rk + 1
14: if xk + 1 does not satisfy inequality constrains (5) then
15: Go to line 9
16: end if
17: Calculate k
18: end for
19: (Start Interior Point method)
20: Set g 0 = 0
22: for k = 1, 2, . . . until k < do
23: if k 1 g k 1 then
24: Calculate Dk
25: Refresh B k and do numerical factorization
26: end if
27: Calculate uk using already obtained Cholesky decomposition
28: Calculate xk + 1
29: if xk + 1 does not satisfy inequality constrains (5) then
30: Go to line 24
31: end if
32: Calculate k
33: end for
5 Matrices Multiplication in Interior Point Method

Approach used in Algorithm 4 allows us to reduce the amount of matrix multi-
plications ADk AT , which is the most expensive operation after Cholesky decom-
position. Additionally, since matrices Dk are diagonal without zero entries, and
A is an incidence matrix, multiplication can be simplied in the following way.
k T
Let us dene B = ADk A , then if i = j and exists t such that Ati = 0 and
k k
Atj = 0, then B ij = Dtk . Otherwise, if i = j and such t does not exist, B ij = 0.
k n k
B ii = j = 1,j = i B ij .
k
Matrix B k is obtained from B by excluding column and row with the same
index as the index of the row excluded from A. Without loss of generality, we
k
can assume that the last row is excluded from A. Explicit computation of B is
unnecessary.
If columns of A are sorted in lexicographic order depending on the row indexes
of nonzero elements, we can say that ji is column index, starting form which
columns cannot have nonzero elements with row indexes less then i. Then index
k
t corresponding to non-diagonal entry Bij is ji plus amount of non-diagonal
k k
nonzero entries of the i-th column of B before Bij . Therefore diagonal entries
are calculated by the following formula

n Dji + 1 1,ji + 1 1 , if column ji + 1 1 has 1
k k
Bii = Bij + nonzero element,

j = i,j=i 0, otherwise.
6 Acceleration by Constant Multiplication

Since constants Si , Hi and Qi are of order 104 , 103 and 102 respectively [2],
in practice it can lead to computational diculties. To improve performance,
scaled vector xs is used instead of x:
xs = x, > 0. (7)
Then problem (1), (3) and (5) is reformulated in the following form:
m
Si |xs |3 Hi s
s s i
F (x ) = 2 xi min, xs IRm ,
i=1
3
1
Axs = Q,

i i i
xsi , i I1 , xsi , i I2 ,

F (xs )
here F s (xs ) = 3 . Numerical experiments show the following. When

= 0.001 = 0.1,
Qi
all constants Si , H i
2 and are approximately of the same order. In this case we
obtain essential acceleration in computations, that can be seen in the Table 1.
7 Combined Method for Constrained Problem

As it can be seen from Table 2, Newton Method works much faster, then Interior
Point Method, but it cannot work with inequality constraints (5) eciently.
Therefore in the case, when inequality constraints are present, the following
scheme is used. Firstly Newton Method is used for solving (1) and (3). Its solution
is projected on the set
= {x | xi i1 , i I1 , xi [i + i , i i ], i = i2 (i i ), i I2 }, (8)
here i1 > 0, i I1 , i2 > 0, i I2 . This set is chosen so that any its point
satises strictly (5). Obtained projection is taken as initial for interior point
search and Interior Point Method. Formalization of the algorithm has form 5.
Results of the algorithm work are presented in Table 3, its comparison with usage
of Interior Point Method is given in Table 4.
Algorithm 5. Combined method

1: Run Newton Method for the problem (1), (3)
2: Do ll-in reduction and symbolic factorization for the matrix AAT
3: Run Algorithm 4 without ll-in reduction and symbolic factorization starting form
the closest point to the one obtained by the Newton Method, that belongs to
(8).
8 Numerical Results
The algorithms were coded in C++. Results of numerical experiments are given
in Tables 1, 2, 3 and 4. Computations were made in PC with Intel Core i7 /
2.4 GHz / 16 GB.
Here the following notations are used: n amount of nodes in system (rows
in A), m amount of edges in system (columns in A), column Scaling describes
whether x is scaled according to (7) or not, Iter. number of iterations, CG
iter. average among all Newton Method steps number of iterations for Con-
jugate Gradient, IPS iter number of iterations of interior (and feasible) point
search (Algorithm 2), Time time in seconds, NZ (%) number of nonzero
entries relative to all entries in the matrices A and Lk , Lk Cholesky matrix,
obtained from B k (nonzero structures of Lk and B k is always same on any iter-
ation, since only diagonal matrix Dk changes), Bck amount of calculations of
values B k and numerical factorizations, Total time full computational time.
In Table 1 problem without inequality constrains (5) is considered. Therefore
Newton Method by itself is sucient to nd optimal solution (theoretically it
can fail because of matrix being indenite or possibly singular, but such cases
did not happen in numerical experiments). Additionally Cholesky factorization
can be done only once, due to the fact that matrices Dk and B k are same for
Table 1. Problem without inequality constraints (5) with and without scaling
Matrix size Scaling Newton Method Interior Point Method

n m CG iter. Iter. Time IPS iter. Iter. Time
1000 1500 No 1327 5 0.16 11 17 0.1
Yes 338 5 0.1 10 6 0.1
5000 7500 No 1406 6 1.3 11 19 2
Yes 398 5 0.35 9 19 2
10000 15000 No 1594 6 2.9 11 22 10.5
Yes 574 7 1.2 9 19 10.1
Table 2. Problem without inequality constraints (5). Newton Method and Dikin Inte-
rior Point Method
Matrix size NZ(%) Newton Method Interior Point Method

N M A Lk CG iter. Iter. Time IPS iter. Iter. Time
1000 1500 0.2 7.6 338 5 0.1 10 6 0.1
5000 7500 0.04 7.3 398 5 0.4 10 6 1.4
10000 15000 0.02 7.3 396 6 0.8 10 7 10
20000 30000 0.01 7.2 419 6 1.7 10 8 73
30000 45000 0.0067 7.1 414 6 2.6 10 8 225
50000 75000 0.004 7.9 443 6 5.2 10 8 1254
100000 150000 0.002 5.6 479 6 14 10 10 6220
1000000 1500000 0.0002 837 6 327 >10000
all iterations. As can be seen, scaling (7) allows to reduce number of iterations
for both Newton Method and Interior Point Method.
In Table 2 Newton Method and Interior Point Method with scaling are com-
pared. As can be seen, Interior Point Method works much slower, because density
of Lk is higher, than density of A, therefore Cholesky decomposition together
with solution of triangular system with Lk require more computations, than
computations with A.
In Table 3 problem (1), (3) and (5) is considered, combined method 5 is
used. Firstly Newton Method is applied to nd solution of problem (1) and (3).
Then obtained point is projected to the shrunk area, as described in Sect. 7
(here i1 = 10, i2 = 0.3). Since in interior point search and in Interior Point
Method matrix B k changes every several iterations, amount of calculations of
B k is presented in column B k calc. for interior point search and for Interior
Point Method.
In Table 4 problem (1), (3) and (5) is considered. Combined method 5 is
compared with usage of Interior Point Method only. As can be seen, with growth
of the problem size, performance dierence increases in favor of method 5.
Table 3. Problem with inequality constraints (5)
Matrix size Newton Method Interior Point Method Total time

IPS
n m CG iter. Iter. Time Iter. Bck Iter. Bck Time
1000 1500 478 10 0.3 10 1 30 2 0.2 0.5
5000 7500 1383 12 2.5 10 1 110 3 7.8 8.2
10000 15000 1476 12 5.4 10 1 737 6 85 95
20000 30000 2631 12 20 10 1 954 7 693 713
30000 45000 3776 12 46 10 1 8351 14 4667 4713
Table 4. Problem with inequality constraints (5). Solution with Newton method and
without it
Matrix size Newton Method Interior Point Method Total time

IPS
n m Iter. B k calc. Iter. B k calc.
1000 1500 Used 10 1 30 2 0.5
Not used 11 2 36 2 0.2
5000 7500 Used 10 1 110 3 8.2
Not used 20 3 61 3 7.9
10000 15000 Used 10 1 737 6 95
Not used 183 3 1199 6 111
20000 30000 Used 10 1 954 7 713
Not used 18 3 3608 6 1198
9 Conclusion
A method, aimed to solve high dimensional ow distribution problem is pre-

sented in this work. Combination of Newton Method and Dikin Interior Point
Method based on sparse matrices techniques is used, to provide solution for
such problems. Numerical experiments show, that it allows to obtain solution
for systems consisting of 10000 30000 nodes in reasonable time on a personal
computer. Additionally presented algorithm provides opportunity for parallel
computations.
Main advantage of the suggested combination of Newton Method and Interior
Point Method consists in the following. This combination grants more precise
solution, than Interior Point Method alone. Since Newton Method works fast, its
usage worsen computational time negligibly. Reduced running time is obtained
due to the good starting point for Interior Point Method provided by Newton
Method.
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Hierarchical Clustering and Multilevel
Refinement for the Bike-Sharing Station
Planning Problem
Christian Kloimullner(B) and Gunther R. Raidl
Institute of Computer Graphics and Algorithms, TU Wien,

Favoritenstrae 911/1861, 1040 Vienna, Austria
{kloimuellner,raidl}@ac.tuwien.ac.at
Abstract. We investigate the Bike-Sharing Station Planning Problem

(BSSPP). A bike-sharing system consists of a set of rental stations, each
with a certain number of parking slots, distributed over a geographical
region. Customers can rent available bikes at any station and return them
at any other station with free parking slots. The initial decision process
where to build stations of which size or how to extend an existing system
by new stations and/or changing existing station congurations is crucial
as it actually determines the satisable customer demand, costs, as well
as the rebalancing eort arising by the need to regularly move bikes from
some stations tending to run full to stations tending to run empty. We
consider as objective the maximization of the satised customer demand
under budget constraints for xed and variable costs, including the costs
for rebalancing. As bike-sharing stations are usually implemented within
larger cities and the potential station locations are manifold, the size
of practical instances of the underlying optimization problem is rather
large, which makes a manual decision process a hardly comprehensible
and understandable task but also a computational optimization very
challenging. We therefore propose to state the BSSPP on the basis of
a hierarchical clustering of the considered underlying geographical cells
with potential customers and possible stations. In this way the estimated
existing demand can be more compactly expressed by a relatively sparse
weighted graph instead of a complete matrix with mostly small non-zero
entries. For this advanced problem formulation we describe an ecient
linear programming approach for evaluating candidate solutions, and for
solving the problem a rst multilevel renement heuristic based on mixed
integer linear programming. Our experiments show that it is possible
to approach instances with up to 2000 geographical cells in reasonable
computation times.
Keywords: Bike-Sharing Station Planning Problem Hierarchical clus-

tering Multilevel renement Facility location problem
1 Introduction
Many large cities around the world have already built bike sharing systems
(BSS), and many more are considering to introduce one or extend an existing
https://doi.org/10.1007/978-3-319-69404-7_11
Hierarchical Clustering and Multilevel Renement for the BSSPP 151
one. These systems consist of rental stations around the city or a certain part
of it where customers can rent and return bikes. A rental station has a specic
number of parking slots where a bike can be taken from or returned to. On the
contrary to bike-rental systems, BSSs encourage a short-term usage of bikes. As
bikes are typically returned at a dierent station than they have been taken
from, a need for active rebalancing arises as the demand for bikes to rent and
parking slots to return bikes is not equally distributed among the stations.
Finding a good combination of station locations and building these stations
in the right size is crucial when planning a BSS as these stations obviously
directly determine the satised customer demand in terms of bike trips, the aris-
ing rebalancing eort, and the resulting xed and variable costs. Stations close
to public transport, business parks, or large housing developments will likely face
a high demand whereas stations in sparser inhabited areas will probably face a
lower demand. However, also the station density and connectedness of the actual
regions to be covered play crucial roles. Some solitary station that is far from
any other station will most likely not fulll much demand. Moreover, a clever
choice of station locations might also exploit the natural demands and customer
ows in order to keep the rebalancing eort and associated costs reasonable.
As BSSs are usually implemented in rather large cities the problem of nd-
ing optimal locations for rental stations and sizing these stations appropriately
is challenging and manually hardly comprehensible. Thus, there is the need for
computational techniques supporting this decision-making. Besides xed costs
for building the system, an integrated approach should also estimate mainte-
nance and rebalancing costs over a certain time horizon such that overall costs
for the operator can be approximated more precisely. It is further important to
consider the customer demands in a time-dependent way because there usually
exists a morning peak and an afternoon peak which is due to commuters, people
going to work, and students. Between these peaks, the demand of the system is
usually a bit lower. We refer to this problem as Bike Sharing Station Planning
Problem (BSSPP). The objective we consider here is to determine for a specied
total-cost budget and a separate xed-cost budget a selection of locations where
rental stations of an also to be determined size should be erected in order to
maximize the actually fullled customer demand.
In this work, we rst concentrate on how to eciently model the BSSPP such
that we can also deal with very large instances with thousands of considered
geographical cells for customers and potential station locations. To this end we
propose to utilize a hierarchical clustering to express the estimated potential
customer demand on it. We will then describe a linear programming (LP) based
method to evaluate candidate solutions, and nally present a rst novel multilevel
renement heuristic (MLR), based on mixed integer linear programming (MIP),
to approach the optimization problem.
In Sect. 2 we discuss related work. Section 3 denes the BSSPP formally, also
introducing the hierarchical clustering. Sections 3.3 and 3.4 describe LP models
for determining the actually fullled customer demands for a candidate solution
and estimating the required rebalancing eort, respectively. The MLR is then
152 C. Kloimullner and G.R. Raidl
described in Sect. 4. First computational results on randomly generated instances

are shown in Sect. 5, and nally, conclusions are drawn in Sect. 6.
2 Related Work
There already exists some work which tries to nd optimal station locations for
BSSs, although mostly considering dierent aspects. To the best of our knowl-
edge, Yang et al. [12] were the rst who considered the problem in 2010. They
relate the problem to hub location problems, a special variant of the well-known
facility location problem, and propose a mathematical model for it. The con-
sidered objective is to minimize the walking distance by prospective customers,
xed costs, and, a penalty for uncovered demands. The authors solve the problem
by a heuristic approach in which a rst part of the algorithm tries to identify the
location of rental stations and a second, inner part tries to nd shortest paths
between origin and destination pairs. The authors illustrate their approach by a
small example consisting of 11 candidate cells for bike stations.
Lin et al. [6] propose a mixed integer non-linear programming model and
solve a small example instance with 11 candidate stations by the commercial
solver LINGO, and furthermore provide a sensitivity analysis. Martinez et al. [8]
develop approaches for a case study within Lisbon having 565 prospective can-
didate stations. They propose a hybrid approach consisting of a heuristic part
utilizing a mixed integer linear programming (MIP) formulation. Locations as
well as the eet dimension are optimized, e-bikes are also considered, and rebal-
ancing requirements are estimated.
Lin et al. [7] propose a heuristic algorithm for solving the hub location inven-
tory problem arising in BSSPP. They do not only optimize station locations
but their algorithm also identies where to build bike lanes. As a subproblem
they have to determine the travel patterns of the customers, i.e., solve a ow
problem for a given conguration. They illustrate their approach on a small
example consisting of 11 candidate locations for stations. Saharidis et al. [9] pro-
pose a MIP formulation which minimizes unmet demands and walking distance
for prospective customers. They test their approach in a case study for the city
center of Athens having 50 candidate cells for stations. Chen et al. [1] provide a
mathematical non-linear programming model and solve the problem utilizing an
improved immune algorithm. They dene three dierent types of rental stations
depending on their location (e.g., near a metro station, supermarkets). Their
aim is that stations in the residential area have enough bikes available such that
the morning peak can be managed and that stations near metro lines or impor-
tant places have enough free parking slots available to manage incoming bikes
during the morning peak. They provide a case study for a particular metro line
of Nianjing city including 10 district stations and 31 residential stations. In [2]
Chen and Sun aim at satisfying a given demand and minimizing travel times
of the users. The authors propose an integer programming model which they
solve with the LINGO solver. A computational analysis is provided on a small
example. Frade et al. [3] describe an approach for a case study of the city of
Coimbra, Portugal. They present a compact MIP model which they solve using
the XPRESS solver. Their objective is to maximize the demand covered by the
BSS under budget constraints. They also include the net revenue in their mathe-
matical model which reduces the costs incurred by building the BSS. Their single
test instance consists only of 29 cells or trac zones, how they call it. Hu et al. [5]
also present a case study for a BSS along a metro line. They aim at minimizing
total costs incurred by building particular BSS stations. In their computational
study they consider three scenarios, each consisting of ten possible station can-
didates. They solve the proposed MIP model by the LINGO solver. Last but
not least, Gavalas et al. [4] summarized diverse algorithmic approaches for the
design and management of vehicle-sharing systems.
We conclude that all previous works on computational optimization
approaches for designing BSS only consider rather small scenarios. Most pre-
vious work accomplishes the optimization with compact mathematical models
that are directly solved by a MIP solver. Such methods, however, are clearly
unsuited for tackling large realistic scenarios of cities with up to 2000 cells or
more. In the following, we therefore propose a novel multilevel renement heuris-
tic based on a hierarchical clustering of the demand data.
3 Problem Formalization
The considered geographical area is partitioned into cells. Let S be the set of
cells where a BSS station may potentially be located (station cells), and let V be
the set of cells where some positive travel demand (outgoing, ingoing, or both)
from prospective customers of the BSS exists (customer cells).
To handle such a large number of cells eectively, we consider a hierarchical
abstraction as crucial in order to represent and model the further data in a mean-
ingful and relatively compact form. To this end, we are expecting a hierarchical
clustering of all customer cells V as input.
This hierarchical clustering is given in the form of a rooted tree with the
inner nodes corresponding to clusters and the leafs corresponding to the cells.
All cells have the same depth which is equal to the height of the tree, denoted by
h. Let C = C0 . . . Ch be the set of all tree nodes, with Cd corresponding to
the subset of nodes at depth d = 0, . . . , h. C0 = {0} contains only the root node
0 representing the single cluster with all cells, while Ch = V . Let super(p) C
be the immediate predecessor (parent cluster) of some node p C \ C0 and
sub(p) C be the set of immediate successors (children) of a cluster p C \ Ch .
As the travel demand of potential users varies over time we are given a (small)
set of periods T = {1, . . . , } for a typical day for which the planning shall
be done. The estimated existing travel demand occurring in each period t T
from/to any cell v V is given by a weighted directed graph Gt = (C t , At ).
All relevant outgoing travel demand at a cell v is represented by outgoing arcs
(v, p) At with p C and corresponding values (weights) dtv,p > 0, i.e., (v, p)
represents all expected trips from v to any cell represented by p in period t that
might ideally be satised, and dtv,p indicates the expected number of these trips.
Moreover, for each time period t T we are given its duration denoted by tperiod
and we are given a global parameter rent which denes the average duration of
a single trip performed by some user of the BSS.
The following conditions must hold to keep this graph as compact and mean-
ingful as possible: the target node p of an arc (v, p) must not be a predecessor
of v in the cluster tree. Self-loops (v, v), however, are allowed and important
to model demand where the destination corresponds to the origin, arcs repre-
senting a neglectable demand, i.e., below a certain threshold, shall be avoided.
Consequently, if there is an arc (v, p) no further arc (v, q) is allowed to any node
q being a successor or a predecessor of p.
All estimated ingoing travel demand for each cell v V is given corre-
spondingly by arcs (p, v) At with p C with demand values dtp,v 0, and
corresponding conditions must hold.
Furthermore, it is an important property, that ingoing and outgoing demands
have to be consistent: Let us denote by V (p) the subset of all cells from V
contained in cluster p C, i.e., the leafs of the subtree rooted in p, and by
C(p) the subset of all the nodes q C that are part of the subtree rooted in p,
including p and V (p). For any p C \ V it must hold that

dtv,q dtq,v (1)
(v,q)At |vV (p),qC(p) (q,v)At |qC(p),vV (p)
and
dtq,v dtv,q . (2)
(q,v)At |qC(p),vV (p) (v,q)At |vV (p),qC(p)
Condition (1) ensures that the total demand originating at the leafs of the subtree
rooted at p and leading to a destination outside of the tree is never less than
the total ingoing demand at all the cells outside the tree originating from some
cluster inside the tree. Condition (2) provides a symmetric condition for the
total ingoing demand at all the leafs of the tree. Furthermore, for the root node
p = 0, inequalities (1) and (2) must hold with equality.
For each customer cell v V , we are given a (typically small) set S(v) S
of station cells in the vicinity by which vs demand may be (partly) fullled.
Furthermore, let av,s (0, 1], v V, s S(v), be an attractiveness value
indicating the expected proportion of demand from v (ingoing as well as out-
going) that can at most be fullled with a suciently sized station at s. These
attractiveness values will be determined primarily based on the walking distances
among the stations (the value will typically roughly exponentially decrease with
the distance), but can be in general an arbitrary distance decay model. If there
is a one-to-one correspondence of cells in V and S, for each v V , v S(v),
av,v = 1 will typically hold.
For the costs of building a station we consider here only a (strongly) sim-
plied linear model, but we distinguish xed costs for building the station and
initially buying the bikes, variable costs for maintaining the station and the
respective bikes, and costs for performing the rebalancing. Let bfix and bvar be
the average xed and variable costs per bike slot, and let breb be the average
costs for rebalancing one bike per day over the whole planning horizon. The xed
costs for a station in cell s S with xs slots are then xcost(s) = bfix xs and
the total costs are totalcost(s) = bfix xs + bvar xs + breb Qx (s), where Qx (s)
denotes an estimation for the number of bikes that need to be redistributed from
station s to some other station. We assume here that the size of each station,
i.e., the number of its slots, can be freely chosen from 0 (i.e., no station is built)
up to some maximum cell-dependent capacity zs N. The determination of the
rebalancing eort for a given candidate solution will be described in Sect. 3.4.
We remark that this cost model only is a rst very rough estimate. Considering
location dependent costs, costs for a station to be built that are independent of
the number of slots, and a more restricted selection of station sizes is left for
future research.
tot
We assume that a total budget Bmax is given as well as a budget for only the
fix tot
sum of all xed costs Bmax < Bmax , and both must not be exceeded in a feasible
solution.
3.1 Solution Representation
A solution x = {xs N | s S} assigns each station cell s S an amount of

parking slots to be built, possibly also 0 which would mean that no station is
going to be built in cell s.
3.2 Objective
The goal is to maximize the expected total number of journeys in the system,
i.e., the total demand that actually can be fullled at each day over all time
tot fix
periods, considering the available budgets Bmax and Bmax .
Let D(x, t) be the total demand fullled by solution x in time period t T ,
and let Qx (s) be the required rebalancing eort arising at each station s S |
xs = 0 in terms of the number of bikes to be moved to some other station. The
calculation of these values will be considered separately in Sects. 3.3 and 3.4.
The BSSPP can then be stated as the following MIP.

max D(x, t) (3)
tT

(bfix xs + bvar xs + breb Qx (s)) Bmax
tot
(4)
sS

bfix xs Bmax
fix
(5)
sS
xs {0, . . . , zs } sS (6)
Inequality (4) calculates the total costs over all stations and ensures that the
total budget is not exceeded, while inequality (5) restricts only the xed costs
over all stations by the respective budget.
3.3 Calculation of Fulfilled Customer Demand
To determine the overall fullled demand for a specic, given solution x and
a certain time slot t T , we rst make the following local denitions. Let
S = {s S | xs = 0} correspond to the set of cells where a station actually
is located, V = {v V | S(v) S = } be the set of customer cells whose
demand can possibly (partly) be fullled as at least one station exists in the
neighborhood. Moreover, let C = {p C | V (p)V = } be the set of all nodes
in the hierarchical clustering representing relevant customer cells, i.e., cells whose
demand can possibly be fullled. The set S (v) = S(v) V , v V refers to
the existing stations that might fulll part of vs demand, and V (p) = V (p)
V , p C denotes the existing customer cells contained in cluster p. C (p)
refers to the subset of all the nodes q C that are part of the subtree rooted at
p, including p and V (p), and G = (C , A ) with A = {(p, q) At | p, q C } is
then the correspondingly reduced demand graph.
In the following we use variables u, v, w for referencing customer cells in V ,
variables p, q for referencing cluster nodes in C (which might possibly also be
customer cells), variable s for station cells in S , and , for arbitrary nodes in
C S.
We further dene for each arc in A corresponding to a specic demand an
individual ow network depending on the kind of the arc:
Arcs (u, v) A with u, v V , including the case u = v:

Gu,v
f = (Vfu,v , Au,v
f ) with node set Vf
u,v
= {u} S (u) S (v) {v} and arc
set Af = ({u} S (u)) (S (u) S (v)) (S (v) {v}).
u,v
Arcs (v, p) A with v V , p C \ V :

Gv,p
f = (Vfv,p , Av,p
v,p
= {v} S (v) {p} and arc set
v,p
Af = ({v} S (v)) (S (v) {p}).
Arcs (p, v) A with p C \ V , v V :
Gp,v
f = (Vfp,v , Ap,v
p,v
= {p} S (v) {v} and arc set
p,v
Af = ({p} S (v)) (S (v) {v}).
All arcs (, ) Ap,q f of all ow networks have associated correspond-

p,q
ing ow variables 0 f, dtp,q . The fullled demands can be modeled
within these networks as maximum ows. Furthermore, we utilize variables
Hpin , Hpout p C \ V , for the total inow/outow at all customer cells V (p)
originating at/targeted to cluster nodes from outside cluster p, i.e., C \C (p)\V .
Variables Fpin , Fpout , p C \ V , represent the total ingoing/outgoing ows at
all cluster nodes q within cluster p originating at/targeted to customer cells out-
side cluster p, i.e., V \ V (p), respectively. The ow variables, however, depend
on each other and the stations capacities. A weighting factor is used to adjust
the number of trips which can be performed in time period t by using only a
single bike. The following LP is used to compute the total satised demand
v,p
D(x, t) = max fv,s (7)
(v,p)A |vV (v,s)Av,p
f
v,p t
s.t. fv,s dv,p (v, p) A | v V (8)
v,p
(v,s)A
f
p,v t
fs,v dp,v (p, v) A | v V (9)
p,v
(s,v)A
f
u,v
u,v
fu,s = fs,s (u, v) A | u, v V , (10)
s S (v)
s S (u)
u,v u,v
fs ,s = fs,v (u, v) A | u, v V , (11)
s S (u)
s S (v)
v,p v,p
fv,s = fs,p (v, p) A | v V , (12)

p C \ V , s S (v)
p,v p,v
fp,s = fs,v (p, v) A | v V , (13)

p C \ V , s S (v)
p,q
xs f,s sS (14)
(p,q)A (,s)Ap,q
f
p,q
fs,
(p,q)A (s,)Ap,q
f

rent (p,q)A (,s)A
p,q
p,q
f,s
f

tperiod
p,q
xs f,s sS (15)
(p,q)A (,s)Ap,q
f
p,q
fs,
(p,q)A (s,)Ap,q
f

rent (p,q)A (s,)A
p,q
p,q
fs,
f
+
tperiod
in

Hp = pC \V (16)
(q,v)A |qC (p)V ,vV (p)
q,v
fs,q
q,v
(s,q)A
f
in

Fp = pC \V (17)
(v,q)A |vV (p),qC (p)\V
v,p
fs,q
v,q
(s,q)A
f
Hpin Fpin p C \ V \ {0} (18)

H0in = F0in (19)

Hpout = pC \V (20)
(v,q)A |vV (p),qC (p)V

q,v
fq,s
(q,s)Aq,v
f

Fpout = p C \ V (21)
(q,v)A |qC (p)\V ,vV (p)

q,v
fq,s
(p,s)Aq,v
f
Hpout Fpout p C \ V \ {0} (22)

H0out = F0out (23)

0 v,p
fv,s av,s dtv,p (v, p) A | v V , (24)
(v, s) Av,p
f
0 fs,v
p,v
as,v dtp,v (p, v) A | v V , (25)
(s, v) Ap,v
f
p,q
0 f, dtp,q (p, q) A , (26)
(, ) Ap,q
f | , V
Fpin , Fpout 0 p C \ V (27)

Hpin , Hpout 0 pC \V (28)
Objective function (7) maximizes the total outgoing ow over all v V , i.e.,
the fullled demand. Note that this also corresponds to the total ingoing ow
over all v. Inequalities (8) limit the total ow leaving v V , for each demand
(v, p) A | v V to dtv,p . Inequalities (9) do the same w.r.t. ingoing demands.
Equalities (10) and (11) provide the ow conservation at source and destination
stations s for (u, v) A with u, v V . Equalities (12) provide the ow conser-
vation at the source station in case of an arc (v, p) A towards a cluster node
p, while (13) provide the ow conservation at the destination station in case
of an arc (p, v) A originating at a cluster node p. Inequalities (14) and (15)
provide the capacity limitations at each station v V . It is the accumulated
demand occurring at the particular station including a compensation term for
large values of ingoing as well as outgoing demand. The fraction tperiod / rent
represents the number of trips which can ideally be performed in period t using
a single bike. The weighting factor is used to adjust this value such that it
better reects reality as the bike trips are not likely to be performed optimally
with respect to the distribution over the whole time period in real world. Equal-
ities (16) compute the total outgoing ow for the leafs of the subtree rooted at
p to any cluster which is not part of the subtree rooted at p. Equalities (17)
compute the total ingoing ow for each cluster node p by considering the ingo-
ing ow from any v V for which p is not a predecessor to every cluster of
the subtree rooted at p. Inequalities (18) ensure that there must not be more
ingoing ow to clusters of the subtree rooted at p as there is outgoing ow from
the leafs contained in the subtree rooted at p. Equality (19) ensures that at the
top level, i.e., at the root node 0, the outgoing ow from leaf nodes to cluster
nodes and the ingoing ow from cluster nodes to leaf nodes is balanced, i.e., the
same amount. Inequalities (21)(23) state the corresponding constraints for the
outgoing ow instead of the ingoing ow. Equations (24) and (25) provide the
domain denitions for the ow variables from/to a cell v to/from a neighboring
station s by considering the demand weighted by factor av,s . For all remaining
ow variables, (26) provide the domain denitions based on the demands. The
remaining variables are just restricted to be non-negative in (27) and (28).
3.4 Calculation of Rebalancing Costs

We state an LP for minimizing the total rebalancing eort over all time periods
T at each station s S by choosing an appropriate initial ll level for each
period, ensuring that the whole prospective customer demand is fullled. We
estimate the rebalancing eort by considering the necessary changes in the ll
levels inbetween the time periods. The LP uses the following decision variables.
By yt,s we refer to the initial ll level of station s S at the beginning of time

period t T , and by rt,s+
and rt,s we denote the number of bikes which need to
be delivered to, respectively picked up from, station s S at the end of period
t T to achieve the ll levels yt+1,s (or y1,s in case of t = ).
acc
The accumulated demand Dt,v can be calculated by utilizing the solution of
the previous model from Sect. 3.3, c.f. inequalities (14) and (15). The following
LP is solved for each station s S | xs = 0 independently. For station cells
s S \ S , i.e., where no station is actually built in solution x, Qx (s) = 0.

+
Qx (s) = min rt,s + rt,s (29)
tT
s.t. +
yt,s + rt,s Dt,s
acc
tT (30)

xs yt,s + Dt,s
rt,s acc
tT (31)

yt+1,s = yt,s Dt,s
acc +
+ rt,s rt,s t T \ { } (32)

y1,s = y,s D,s
acc +
+ r,s r,s (33)
0 yt,s xs tT (34)
0 +
rt,s acc
Dt,s tT (35)

0 rt,s Dt,s
acc
tT (36)
Objective function (29) minimizes the number of rebalanced bikes, i.e., number
+
of bikes that have to be delivered rt,s and number of bikes that have to be

picked up rt,s . Inequalities (30) compute the number of bikes that have to
be delivered to the corresponding station in order to meet the given demand.

Inequalities (31) compute the number of bikes that have to be picked up from
the corresponding station in order to meet the given demand. Inequalities (32)
state a recursion in order to compute the ll level for the next time period.
Inequalities (33) state that for each station the ll level for the next day has
to be again the initial ll level of the rst period. Inequalities (34)(36) are the
domain denitions for the number of bikes to be moved and the ll level for each
time period.
4 Multilevel Refinement Approach
Clearly, practical instances of the problem are far too large to be approached
by a direct exact MIP approach. However, also basic constructive techniques
or metaheuristics with simple, classical neighborhoods are unlikely to yield rea-
sonable results when making decisions on a low level without considering crucial
relationships on higher abstraction levels, i.e., a more global view. Classical local
search techniques on the natural variable domains concerning decisions for indi-
vidual stations may only ne-tune a solution but are hardly able to overcome
bad solutions in which larger regions need to be either supplied with new sta-
tions or where many stations need to be removed. We therefore have the strong
need of some technique that exploits also a higher-level view, deciding for larger
areas about the supply of stations in principle. Multilevel renement strategies
can provide this point-of-view.
In multilevel renement strategies [11] the whole problem is iteratively coars-
ened (aggregated) until a certain problem size is reached that can be reasonably
handled by some exact or heuristic optimization technique. After obtaining a
solution at this highest abstraction level, the solution is iteratively extended to
the previous lower level problem instance and possibly rened by some local
search, until a solution to the original problem at the lowest level, i.e., the orig-
inal problem instance, is obtained. For a general discussion and the generic
framework we refer to the work of Walshaw [10].
To apply multilevel renement to BSSPP we essentially have to decide how
to realize the procedures for coarsening an instance for the next higher level,
solving a reasonably small instance, and extending a solution to a solution at
the next lower level. In the following, we denote all problem instance data at
level l by an additional superscript l. By Pl we generally refer to the problem at
level l of the MLR algorithm described here.
4.1 Coarsening
We have to derive the more abstract problem instance Pl+1 from a given instance
Pl . Naturally, we can exploit the already existing customer cell cluster hierar-
chy for the coarsening. Remember that all customer cells appear in the cluster
hierarchy always at the same level. We coarsen the problem by considering the
customer cells and the station cells separately.
Coarsening of Customer Cells. The main strategy for coarsening the customer
cells is to merge cells having the same parent cluster together with their parent.
This means V l+1 = Chl l 1 or simply V l+1 = Chl1 , i.e., each cluster node at
depth h l 1 corresponds to a customer cell at level l + 1 representing the
merged set of customer nodes contained in Chl1 . The hierarchical clustering
of Pl becomes C l+1 = C0 . . . Chl . Remember that we already dened
the function super(p) to return the parent cluster of some node p, and therefore
super(pl ) : C l C l+1 also returns the cluster from C l+1 in which cluster pl C l
is merged into. The new demand graph Gt,l+1 = (C t,l+1 , At,l+1 ) consists of the
t,l+1
arc set A = (pl ,ql )At,l (super(p ), super(q l )). This demand graph may again
l
contain self-loops, but it is still simple, i.e., multiple arcs from At,l may map to
the same single arc in At,l+1 and the respective demand values are merged.
Considering an arc (pl+1 , q l+1 ) At,l+1 , its associated demand is thus

dt,l+1
pl+1 ,q l+1 = dt,l
pl ,q l
. (37)
(pl ,q l )At,l |pl+1 =super(pl ),q l+1 =super(q l )
Note that the conditions for a valid demand graph and valid demand values
stated in inequalities (1) and (2) will still hold when aggregating in this way,
since the total ingoing and outgoing demand at each cluster p C l+1 (including
the demands from and to all existing subnodes) stays the same.
Coarsening of Station Cells. To coarsen the station cells we need to dene a

hierarchical clustering for them as well. For simplicity we assume from now on
that S = V holds, i.e., there is a one-to-one correspondence of considered station
cells and customer cells. This also appears reasonable in a practical setting. We
can then apply the hierarchical clustering dened for the customer cells also to
the station cells. Maximum station capacities for aggregated stations sl+1 S l+1
are naturally calculated by the sum ofthe respective maximum capacities of the
underlying station cells, i.e., zsl+1
l+1 =
l
sl sub(sl+1 ) zsl .
Coarsening of Neighborhoods. A coarsened neighborhood mapping S l+1 (v l+1 )

for each customer cell v l+1 V l+1 and respective attractiveness values avl+1 ,sl+1
for station cells sl+1 S l+1 (v l+1 ) are determined as follows. The neighborhood
mapping is retained as long as the attractiveness value in the coarsened problem
instance does not fall below a certain threshold (0, 1):

S l+1 (v l+1 ) = sl+1 super(S l (v l )) | avl+1 ,sl+1 (38)

v l sub(v l+1 )
with the aggregated attractiveness values being

1 if v l+1 = sl+1
avl+1 ,sl+1 = vl sub(vl+1 ) sl sub(sl+1 )Sl (vl ) (avl ,sl ) (39)
l+1 l+1 if v l+1 = sl+1 .
|sub(v )||sub(s )|
4.2 Initialization
The initial problem becomes coarsened until we reach some level l where it can
be reasonably solved as it is then small enough. In our experiments with binary
clustering trees here we are stopping the coarsening when the clustering tree
has no more than 25 = 32 leaf nodes, or in other words, at a height of ve.
For initializing the solution at the coarsest level we utilize a MIP model. In
this model, the objective stated in Sect. 3.2, the demand calculation for every
time period stated in Sect. 3.3, and the rebalancing LP model stated in Sect. 3.4
are put together. By solving this model we obtain an optimal solution for the
coarsest level, which forms the basis for proceeding with the next step of the
algorithm, the extension to derive step-by-step a more detailed solutions.
4.3 Extension
In the extension step we derive from a solution xl+1 at level l + 1 a solution xl
at level l, i.e., we have to decide for each aggregated station sl+1 S l+1 with
xl+1
sl+1
> 0 slots how they should be realized by the respective underlying station
cells sub(sl+1 ) at level l. We do this in a way so that the globally fullled demand
is again maximized by solving the following MIP.

max D(xl , t) (40)
tT

s.t. bfix xlsl + bvar xlsl + breb Qxl (sl ) Bmax
tot
(41)
sl S l

bfix xsl Bmax
fix
(42)
sl S l

xlsl xl+1
sl+1
sl+1 S l+1 (43)
sl sub(sl+1 )
xlsl {0, . . . , zsl } sl S l (44)
The objective (40) maximizes the total satisable demand. Inequalities (41)
restrict the maximum total budget whereas inequalities (42) restrict the maxi-
mum xed budget. Inequalities (43) are the bounds on the total number of slots
for the station nodes sl sub(sl+1 ). The number of parking slots in each cell xlsl
is restricted by the maximum number of parking slots allowed in this cell (44).
5 Computational Results
For our experiments we created seven dierent benchmark sets1 , each one con-
taining 20 dierent, random instances. We consider instances with 200, 300, 500,
800, 1000, 1500, and 2000 customer cells, where each customer cell is also a pos-
sible location for a station to be built. Customer cells are aligned on a grid in
1
https://www.ac.tuwien.ac.at/les/resources/instances/bsspp/lion17.bz2.
the plane and euclidean distances have been calculated based on which a hier-
archical clustering with the complete-linkage method was computed. Demands
among the leaf nodes were chosen randomly, considering the pairwise distance
between customer cells, and demands below a certain threshold have been aggre-
gated upwards in the clustering tree such that the demand graphs get sparser.
Only cells within 200 m walking distance are considered to be in the vicinity of
a customer cell and respective attractiveness values are chosen randomly but in
correlation with the distances. We set the maximum station size to zs = 40 for
all cells in all test cases. For slot costs we set bfix = 1750 e, and bvar = 1000 e,
which are reasonable estimates in the Vienna area gathered from real BSSs. The
costs for rebalancing a single bike for one day have been estimated with 3 e per
bike and per day. When projecting this cost to the optimization horizon, e.g.,
1 year, we get breb = 365 3 = 1095 e. For coarsening of attractiveness values,
we set the corresponding parameter = 0 and for adjusting the number of trips
which can be performed in a particular time period t T by using only a single
bike we set = 1.2. Each instance contains four time periods which we selected
as follows: 4:30 am to 8:00 am, 8:00 am to 12:00 Noon, 12:00 Noon to 6:15 pm,
and 6:15 pm to 4:30 am. The duration for each time period t T has been set
accordingly and the average trip duration has been set to trent = 10 min.
All algorithms are implemented in C++ and have been compiled with gcc 4.8.
For solving the LPs and MIPs we used Gurobi 7.0. All experiments were executed
as single threads on an Intel Xeon E5540 2.53 GHz Quad Core processor.
Table 1 summarizes obtained results. For every instance set we state the
name containing the number of nodes, the number of dierent instances we
tot
have tested on (#runs), the maximum total budget (Bmax ), and the maximum
fix
xed budget (Bmax ). For the proposed MLR, we list the average objective value
(obj), i.e., the expected fullled demand in terms of the number of journeys, the
average number of coarsening levels (#coarsen), the median time (time), and
the average total costs (totcost) as well as the average xed costs (xcost) for
building the number of slots in the solution. Most importantly, it can be seen
that the proposed MLR scales very well to large instances up to 2000 customer
cells.
Table 1. Results for the multilevel renement heuristic (MLR).
Instance MLR
tot fix [s] totcost [e]
Name #runs Bmax [e] Bmax [e] obj #coarsen time fixcost [e]
BSSPP 200 20 200,000.00 130,000.00 9,651.98 3 46.2 198,000.00 126,000.00
BSSPP 300 20 350,000.00 250,000.00 10,951.79 5 60.8 349,250.00 222,250.00
BSSPP 500 20 500,000.00 350,000.00 16,057.78 6 121.6 497,750.00 316,750.00
BSSPP 800 20 850,000.00 550,000.00 28,862.21 6 263.9 849,750.00 540,750.00
BSSPP 1000 20 1,000,000.00 700,000.00 28,967.58 8 346.7 998,250.00 635,250.00
BSSPP 1500 20 1,500,000.00 1,000,000.00 41,208.19 8 574.5 1,498,475.00 953,575.00
BSSPP 2000 20 2,000,000.00 1,300,000.00 55,892.06 8 803.4 1,999,250.00 1,272,250.00
Average 27,370.22 6.3 912,960.71 580,975.00
6 Conclusion and Future Work

We presented an innovative approach to the BSSPP. Previous work only consid-
ers very small instances and case studies to small parts of a city whereas we aim
at solving more realistic large-scale scenarios arising in large cities. As we have
to cope with thousands of customer cells and potential station cells it is most
fundamental to model the potential demands eciently. To this end, we pro-
posed to use a hierarchical clustering and dening the demand graph on it. This
approach can drastically reduce the data in comparison to a complete demand
matrix with only a very reasonable information loss. Moreover, we provided MIP
formulations to compute the satisable demand by given congurations and to
compute the prospective rebalancing costs. Putting them together under the
objective of maximizing the expected satised total demand and adding further
constraints for complying with given monetary budget constraints, we obtained
a MIP model that solves our denition of the BSSPP exactly. Because this MIP
model can in practice still only be solved for rather small instances, we further
suggested a multilevel renement heuristic utilizing the same hierarchical clus-
tering we are given as input. Using this approach we have shown to be able to
solve instances with up to 2000 nodes in reasonable computation times.
In future work it is important to make the cost model more realistic and to
test on more realistic benchmark instances. In particular, we aim at considering
also xed costs for building a station which are independent of the number of
slots. Furthermore, in practice also only a small, restricted set of dierent station
congurations is possible per station cell. These extensions introduce interesting
research questions especially in relation to the multilevel renement procedure.
Acknowledgements. We thank the LOGISTIKUM Steyr, the Austrian Institute of

Technology, and Rosinak & Partner for the collaboration on this topic. This work is
supported by the Austrian Research Promotion Agency (FFG) under contract 849028.
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Decomposition Descent Method for Limit
Optimization Problems
Igor Konnov(B)
Institute of Computational Mathematics and Information Technologies,

Kazan Federal University, Kremlevskaya St. 18, 420008 Kazan, Russia
konn-igor@ya.ru
http://kpfu.ru
Abstract. We consider a general limit optimization problem whose

goal function need not be smooth in general and only approximation
sequences are known instead of exact values of this function. We suggest
to apply a two-level approach where approximate solutions of a sequence
of mixed variational inequality problems are inserted in the iterative
scheme of a selective decomposition descent method. Its convergence is
attained under coercivity type conditions.
Keywords: Optimization problems Limit problems Non-smooth

functions Mixed variational inequality Decomposition descent
method Coercivity conditions
1 Introduction
We rst consider the general optimization problem, which consists in nding
the minimal value of some function p over the corresponding feasible set X. For
brevity, we write this problem as
min p(x). (1)

xX
Its solution set will be denoted by X and the optimal value of the function by
p , i.e.
p = inf p(x).
xX
In order to develop ecient solution methods for this problem we should exploit
certain additional information about its properties, which are related to some
classes of applications.
In what follows, we denote by Rs the real s-dimensional Euclidean space, all
elements of such spaces being column vectors represented by a lower case Roman
alphabet in boldface, e.g. x. For any vectors x and y of Rs , we denote by x, y
their scalar product, i.e.,
s

x, y = x y = xi yi ,
i=1
https://doi.org/10.1007/978-3-319-69404-7_12
Decomposition Descent Method 167

and by x the Euclidean norm of x, i.e., x = x, x. Next, we dene for
brevity M = {1, . . . , n}, |A| will denote the cardinality
of a nite set A. As
usual, R will denote the set of real numbers, R = R {+}.
Let us consider a partition of the N -dimensional space
RN = RN1 . . . RNn , (2)
i.e.
n

N = Ni ,
i=1

where N = {1, . . . , N }, N = |N |, Ni = |Ni |, and Ni Nj = if i = j.
This means that any point x = (x1 , . . . , xN ) RN is represented by x =
(x1 , . . . , xn ) where xi = (xj )jNi RNi for i M . The simplest case where
ni = 1 for all i M and n = N corresponds to the scalar coordinate partition.
Rather recently, partially decomposable optimization problems were paid sig-
nicant attention due to their various big data applications; see e.g. [13] and
the references therein. In these problems, the cost function and feasible set are
specialized as follows:
p(x) = f (x) + h(x), (3)

n
h(x) = hi (xi ), (4)
i=1
n

X = X1 . . . X n = Xi , (5)
i=1
where f : RN R is a function, which is continuous on X, hi : RNi R

is a convex function, and Xi is a convex set in RNi for i = 1, . . . , n. Note
that the function f : RN R is not supposed to be convex in general. That
is, we have to solve a non-convex and non-dierentiable optimization problem,
which appears too dicult for solution with usual subgradient type methods.
Nevertheless, one can develop ecient coordinate-wise decomposition methods
for nding stationary points of problem (1), (3)(5) for a smooth f ; see e.g.
[36] and the references therein. Then the stationary points can be dened as
solutions of the following mixed variational inequality (MVI for short): Find a
point x X such that
n

f (x ), y x + [hi (yi ) hi (xi )] 0
(6)
i=1
yi Xi , for i = 1, . . . , n.
Here also y = (y1 , . . . , yn ) as above.

We observe that all these solution methods also require exact values of the
cost function and parameters of the feasible set. However, this is often impossible
for big data problems due to the calculation errors and incompleteness of the
168 I. Konnov
necessary information. In addition, the same situation arises if we nd it useful to

replace the initial problem by a sequence of auxiliary ones with better properties.
For instance, we can suppose that f in (3) is a general non-smooth function and
also replace it with a sequence of its smooth approximations. In other words, we
have to develop methods for limit (or non-stationary) problems.
There exist a number of methods for such limit optimization and variational
inequality problems, but they are based essentially upon convexity assump-
tions and restrictive concordance rules for accuracy, approximation, and iter-
ation parameters, which creates serious diculties for their implementation; see
e.g. [79] and the references therein. For instance, if iteration parameters are
dependent of attained accuracy or approximation, they can not be evaluated
properly. Besides, the mutual dependence of these parameters usually leads to
very restrictive control rules and slow convergence.
In this paper, we intend to suggest a descent coordinate-wise decomposition
method for the following problem: Find a point x X such that
n

g G(x ), g , y x + [hi (yi ) hi (xi )] 0
(7)
i=1
yi Xi , for i = 1, . . . , n;
where G : X (RN ) is a point-to-set mapping whose values are considered

as generalized gradient sets of the function f ; cf. (6). Here (A) denotes the
family of all nonempty subsets of a set A. For instance, if f is a locally Lipschitz
function, we can set G to be its Clarke subdierential mapping. If f is a convex
function, we simply set G(x) to be the usual subdierential f (x) of f at x,
then each solution of (7) clearly solves (1), (3)(5). Next, we suppose that only
sequences of approximations are known instead of the exact values of G and h.
In creating the desired solution method for the limit (or non-stationary) prob-
lem (7), we combine the selective descent splitting method from [6] with chang-
ing the tolerance parameters corresponding to a sequence of mixed variational
inequality problems, which does not require evaluation of accuracy of approxi-
mate solutions, and utilization of some coercivity conditions; see e.g. [10]. This
approach allows us to prove convergence without special concordance rules for all
the parameters and tolerances.
2 Auxiliary Problem Properties
Let us consider a partially partitionable optimization problem of the form
min (x) = {(x) + (x)} , (8)

xX
where the function : RN R is smooth on X, but not necessary convex. This

problem will serve as approximation of the basic problem (1), (3)(5). We will use
the same partition (2) of the space RN and x the assumption on the feasible set.
(A1) It holds that (5) where Xi are non-empty, convex, and closed sets in RNi
for i = 1, . . . , n.
Also, we suppose that
n

(x) = i (xi ), (9)
i=1
where i : RNi R is convex and has the non-empty subdierential i (xi ) at

each point xi Xi , for i M . Then each function i is lower semi-continuous,
the function is lower semi-continuous, and
(x) = 1 (x1 ) . . . n (xn ).
So, our problem (8) and (9) is rewritten as
n

min (x) = (x) + i (xi ) . (10)
xX1 ...Xn
i=1
Set g(x) = (x), then

(x)
g(x) = (g1 (x), . . . , gn (x)) , where gi (x) = RNi , i = 1, . . . , n.
xj jNi
Given a point x X, we say that a vector d is feasible for x if x + d X

for some > 0. From the assumptions above it follows that the function is
directionally dierentiable at each point x X, that is, its directional derivative
with respect to any feasible vector d is dened by the formula:
n

(x; d) = g(x), d + (x; d), with (x; d) = max bi , di ;
bi i (xi )
i=1
see e.g. [11].

We recall that a function f : Rs R is said to be coercive on a set D Rs
if {f (uk )} + for any sequence {uk } D, uk . We will in addition
suppose that the function : RN R is coercive on X, then problem (8) and
(9) (or (10)) has a solution.
Problem (10) was considered in particular in [6]. We will utilize some prop-
erties obtained there and the descent splitting method from [6] will serve as a
basic element of the two-level method for the general limit optimization problem.
For this reason, we give some properties from [6] without proofs. We start our
considerations from the optimality condition.
Lemma 1 [6, Proposition 2.1].
(a) Each solution of problem (10) is a solution of the following MVI: Find a
point x X = X1 Xn such that
n
n

gi (x ), yi xi + [i (yi ) i (xi )] 0
(11)
i=1 i=1
yi Xi , for i = 1, . . . , n.
170 I. Konnov
(b) If is convex, then each solution of MVI (11) solves problem (10).
In what follows, we denote by X 0 the solution set of MVI (11) and call it the
set of stationary points of problem (10); cf. (6).
Fix > 0. For each point x X we can dene y(x) = (y1 (x), . . . , yn (x)) X
such that
n
n

gi (x) + (yi (x) xi ), yi yi (x) + [i (yi ) i (yi (x))] 0
(12)
i=1 i=1
yi Xi , for i = 1, . . . , n.
This MVI gives a necessary and sucient optimality condition for the optimiza-
tion problem:
n
min i (x, yi ), (13)
yX1 ...Xn
i=1
where
i (x, yi ) = gi (x), yi + 0.5xi yi 2 + i (yi ) (14)
for i = 1, . . . , n. Under the above assumptions each i (x, ) is strongly convex,
hence problem (13) and (14) (or (12)) has the unique solution y(x), thus dening
the single-valued mapping x y(x). Observe that all the components of y(x)
can be found independently, i.e. (13) and (14) is equivalent to n independent
optimization problems of the form
min i (x, yi ), (15)

yi Xi
for i = 1, . . . , n and yi (x) just solves (15).
Lemma 2 [6, Proposition 2.2].
(a) x = y(x) x X 0 ;
(b) The mapping x y(x) is continuous on X.

n
Set (x) = x y(x), then 2 (x) = 2i (x) where i (x) = xi yi (x).
i=1
From Lemma 2 we conclude that the value (x) can serve as accuracy measure
for MVI (11).
We need also a descent property from [6, Lemma 2.1].
Lemma 3. Take any point x X and an index i M . If

yi (x) xi if s = i,
ds =
0 if s = i;
then
(x; d) yi (x) xi 2 .
Denote by Z+ the set of non-negative integers. Following [6] we describe the

basic algorithm for MVI (11) as follows.
Algorithm (DDS). Input: A point x0 X. Output: A point z. Parameters:
Numbers > 0, > 0, (0, ), (0, 1).
At the k-th iteration, k = 0, 1, . . ., we have a point xk X.
Step 1: Choose an index i M such that i (xk ) , set ik = i,

k ys (xk ) xks if s = ik ,
ds =
0 if s = ik ;
and go to Step 3. Otherwise (i.e. when s (xk ) < for all s M ) go to Step 2.
Step 2: Set z = xk and stop.
Step 3: Determine m as the smallest number in Z+ such that
(xk + m dk ) (xk ) m 2i (xk ), (16)
set k = m , xk+1 = xk + k dk , and k = k + 1. The iteration is complete.

Although its properties are similar to those in [6], we give their proofs here
for more clarity of exposition.
Lemma 4. The line-search procedure in Step 3 is always finite.
Proof. If we suppose that the line-search procedure is innite, then
m ((xk + m dk ) (xk )) > 2i (xk ),
for m , hence, by taking the limit we have (xk ; dk ) 2i (xk ), but

Lemma 3 gives (xk ; dk ) 2i (xk ), hence , a contradiction.

We obtain the main property of the basic cycle.
Proposition 1. The number of iterations in Algorithm (DDS) is finite.
Proof. By construction, we have < (xk ) and (xk+1 ) (xk )

2 k , hence the sequence {xk } is bounded and has limit points, besides,
lim k = 0.
k
Suppose that the sequence {xk } is innite. Since the set M is nite, there is an
index ik = i, which is repeated innitely. Take the corresponding subsequence
{ks }, then, without loss of generality, we can suppose that the subsequence {xks }
converges to a point x, besides, iks (xks ) = dki s , and we have
(ks /)1 ((xks + (ks /)dks ) (xks )) > dki s 2 .
Using the mean value theorem (see e.g. [11, Theorem 2.3.7]), we obtain
giks + tki s , dki s = gks + tks , dks > dki s 2 ,

172 I. Konnov
for some gks = (xks + (ks /)ks dks ), tks (xks + (ks /)ks dks ), ks
(0, 1). By taking the limit s we have
(x) + t, d = gi (x) + ti , di di 2 ,
for some t (x), where

yi (x) xi if s = i,
ds =
0 if s = i;
due to Lemma 2. On the other hand, using Lemma 3 gives
(x) + t, d (x; d) di 2 ,
besides, by construction, we have dki s , hence di > 0 and ,

which is a contradiction.

3 Limit Decomposition Method and its Convergence

We now intend to describe a general iterative method for the limit MVI (7).
First we introduce the approximation assumptions.
(A2) There exists a sequence of continuous mappings Gl : X RN , which

are the gradients of functions fl : RN R, l = 1, 2, . . . , such that the
relations {yl } y and yl X imply {Gl (yl )} g G(y).
(A3) For each i = 1, . . . , n there exists a sequence of convex functions hl,i :
RNi R, such that each of them is subdierentiable on Xi and that the
relations {ul } u and ul Xi imply {hl,i (ul )} hi (u).
Condition (A2) means that the limit set-valued mapping G at any point
is approximated by a sequence of gradients {Gl }. In fact, if G is the Clarke
subdierential of a locally Lipschitz function f , it can be always approximated
by a sequence of gradients within condition (A2); see [12,13]. Observe also
that if there is a subsequence yls X with {yls } y, then (A2) implies
{Gls (yls )} g G(y) and the same is true for (A3). At the same time, the
non-dierentiability of the functions f or h is not obligatory, the main property
is the existence of the approximation sequences indicated in (A2) and (A3).
So, we replace MVI (7) with a sequence of MVIs: Find a point zl X =
X1 . . . Xn such that
n
n

Gl,i (zl ), yi zli + hl,i (yi ) hl,i (zli ) 0
(17)
i=1 i=1
yi Xi , for i = 1, . . . , n;
where we use the partition of Gl which corresponds to that of the space RN , i.e.
Gl (x) = (Gl,1 (x), . . . , Gl,n (x)) , where Gl,i (x) RNi , i = 1, . . . , n.

Similarly, we set
n

hl (x) = hl,i (xi ).
i=1
Since the feasible set X may be unbounded, we introduce also coercivity

conditions.
(C1) For each xed l = 1, 2, . . . , the function fl (x) + hl (x) is coercive on the
set X, that is, {fl (wk ) + hl (wk )} + if {wk } X, wk as
k .
(C2) There exist a number > 0 and a point v X such that for any sequences
{ul } and {dl } satisfying the conditions:
ul X, {ul } +, {dl } 0;
it holds that

lim inf Gl (ul ) + dl , v ul dl + [hl (v) hl (ul dl )] if > 0.
l
Clearly, (C1) gives a custom coercivity condition for each function fl (x) +
hl (x), which provides existence of solutions of each particular problem (17).
Obviously, (C1) holds if X is bounded. At the same time, (C2) gives a similar
coercivity condition for the whole sequence of these problems approximating the
limit MVI (7). It also holds if X is bounded. In the unbounded case (C2) is
weaker than the following coercivity condition:

vul dl 1 Gl (ul ) + dl , v ul dl + [hl (v) hl (ul dl )] as l .
Similar conditions are also usual for penalty type methods; see e.g. [14,15]. We
therefore conclude that conditions (C1) and (C2) are not restrictive.
The whole decomposition method for the non-stationary MVI (7) has a two-
level iteration scheme where each stage of the upper level invokes Algorithm
(DDS) with dierent parameters.
Method (DNS). Choose a point z0 X and a sequence {l } +0.
At the l-th stage, l = 1, 2, . . ., we have a point zl1 X and a number l .
Set
(x) = fl (x), (x) = hl (x),
apply Algorithm (DDS) with x0 = zl1 , = l and obtain a point zl = z as its
output.
We now establish the main convergence result.
Theorem 1. Suppose that assumptions (A1)(A3) and (C1)(C2) are ful-

filled, besides, {l } +0. Then:
(i) problem (17) has a solution;

(ii) the number of iterations at each stage of Method (DNS) is finite;
174 I. Konnov
(iii) the sequence {zl } generated by Method (DNS) has limit points and all these
limit points are solutions of MVI (7);
(iv) if f is convex, then all the limit points of {zl } belong to X .
Proof. We rst observe that (C1) implies that each problem (17) has a solution
since the cost function
(x) = fl (x) + hl (x)
is coercive, hence the set

Xl (x0 ) = y X | (y) (x0 )
is bounded. It follows that the optimization problem
min (x)
xX
has a solution and so is MVI (17) due to Lemma 1. Hence, assertion (i) is true.
Next, from Proposition 1 we now have that assertion (ii) is also true.
By (ii), the sequence {zl } is well-dened and (12) implies
Gl (zl ) + (yl (zl ) zl ), y yl (zl ) + [hl (y) hl (yl (zl ))] 0 y X. (18)
Besides, the stopping rule in Algorithm (DDS) gives

l n zl yl (zl ). (19)
We now proceed to show that {zl } is bounded. Conversely, suppose that

{zl } +. Applying (18) with y = v, we have
0 gl + dl , v zl + [hl (v) hl (zl )].
Here and below, for brevity we set gl = Gl (zl ), zl = yl (zl ), and dl = (y(zl )zl ).
Take a subsequence {ls } such that

lim gls + dls , v zls + [hls (v) hls (zls )]
s

= lim inf gl + dl , v zl + [hl (v) hl (zl )] ,
l
then, by (C2), we have

0 lim gls + dls , v zls + [hls (v) hls (zls )] < 0,
s
a contradiction. Therefore, the sequence {zl } is bounded and has limit points.
Let z be an arbitrary limit point for {zl }, i.e.
z = lim zls .
s
Since zl X, we have z X. It now follows from (A2) that lims gls = g

G(z).
Fix an arbitrary point y X, then, using (18) and (19) and (A3), we have

g, y z + [h(y) h(z)] = lim gls , y zls + [hls (y) hls (zls )]
s

= lim gls + dls , y zls + [hls (y) hls (zls )] 0,
s
therefore z solves MVI (7) and assertion (iii) holds true.

Next, if f is convex, then so is p and each limit point of {zl } belongs to X ,
which gives assertion (iv).

We observe that the above proof implies that MVI (7) has a solution.
4 Modifications and Applications

Method (DNS) admits various modications. In particular, we can take the exact
one-dimensional minimization rule instead of the current Armijo line-search (16)
in Algorithm (DDS), then the assertions of Theorem 1 remain true. Next, if the
function (i.e. each function fl ) is convex, we can replace (16) with the following:

gi (xk + m dk ), dki + m i (xki + m dki ) i (xki ) 1 2i (xk ).
Moreover, if the gradient of the function is Lipschitz continuous, we have an

explicit lower bound for the step-size and utilize the xed step-size version of
Algorithm (DDS), which leads to further reduction of computational expenses.
It can be applied if the partial gradients of is Lipschitz continuous; see [6] for
more details.
We give now only two instances in order to illustrate possible applications.
The rst instance is the linear inverse problem that arises very often in signal
and image processing; see e.g. [16] for more examples. The problem consists in
solving a linear system of equations
Ax = b,
where A is a m n matrix, b is a vector in Rm , whose exact values may be

unknown or admit some noise perturbations. If A A is ill-conditioned, the cus-
tom approach based on the least squares minimization problem
min Ax b2

x
may give very inexact approximations. In order to enhance its properties, one
can utilize a family of regularized problems of the form
min Ax b2 + h(x), (20)

x

n
where h(x) = x2 or h(x) = x1 |xi |, > 0 is a parameter. Note that the
i=1
non-smooth regularization term yields additionally sparse solutions with rather
small number of non-zero components; see e.g. [2,17].
176 I. Konnov
The second instance is the basic machine learning problem, which is called
the linear support vector machine. It consists in nding the optimal partition
of the feature space Rn by using some given training sequence xi , i = 1, . . . , l
where each point xi has a binary label yi {1, +1} indicating the class. We
have to nd a separating hyperplane. Usually, its parameters are found from the
solution of the optimization problem
l

minn (1/p)wpp + C L(w, xi ; yi ), (21)
wR
i=1
where L is a loss function and C > 0 is a penalty parameter. The usual choice
is L(z; y) = max{0; 1 yz} and p is either 1 or 2; see e.g. [1,5] for more details.
Observe that the data of the observation points xi can be again inexact or even
non-stationary.
Next, taking p = 2, we can rewrite this problem as
l

min 0.5w2 + C i ,
w,
i=1
subject to
1 yi w, xi i , i 0, i = 1, . . . , l.
Its dual has the quadratic programming format:
l
l
l
max i 0.5 (s ys )(t yt )xs , xt . (22)
0i C,i=1,...,l
i=1 s=1 t=1
Observe that all these problems fall into format (1), (3)(5) and that they can
be treated as limit problems.
5 Computational Experiments
In order to evaluate the computational properties of the proposed method we
carried out preliminary series of test experiments. For simplicity, we took only
unconstrained test problems of form (1), (3)(5) where X = RN with the single-
dimensional (coordinate) partition of the space, i.e., set N = n and Ni = 1 for
i = 1, . . . , n. In all the experiments, we took the limit function f to be convex
and quadratic, namely,
f (x) = 0.5Ax b2 + 0.5x2 ,
where A was an n n matrix, b a xed vector whose elements were dened by

trigonometric functions, whereas the limit function h was dened either as zero or
a non-smooth and convex one. In view of examples (20)(22) from Sect. 4, these
limit problems were approximated by the perturbed sequence of the following
optimization problems
min {fl (x) + h(x)} , (23)
xX
i.e. we utilized only the perturbation for f , where
fl (x) = 0.5A(l )x b(l )2 + 0.5x2 ,

aij () = aij + 0.1 sin(ij) and bi () = bi + 0.2 sin(i), for i, j = 1, . . . , n.
The main goal was to compare (DNS) with the usual (splitting) gradient descent
method (GNS for short); see [18]. It calculates all the components for the direc-
tion nding procedure. Both the methods used the same line-search strategy and
were applied sequentially to each non-stationary problem (23) with the following
rule l+1 = l for changing the perturbation. In (GNS), this change occurs after
satisfying the inequality
l (x) = x yl (x) l ,
where yl (x) is a unique solution of the problem

min fl (x), y + 0.5x y2 + h(y) .
y
In both the methods, we chose the rule l+1 = l with = 0.5. Similarly, for
the limit problem, we set
(x) = x y(x),
where y(x) is a unique solution of the problem

min f (x), y + 0.5x y2 + h(y) .
y
We took (xk ) as accuracy measure for solving the limit problem, chose the
accuracy 0.1, took the same starting point zj0 = j| sin(j)| for j = 1, . . . , n, and
set = 1 for both the methods. The methods were implemented in Delphi with
double precision arithmetic.
In the rst two series, we set h 0 and took versions with exact line-search. In
the rst series, we took the elements aij = sin(i/j) cos(ij) and bi = (1/i) sin(i).
The results are given in Table 1. In the second series, we took the elements aij =
1/(i + j) + 2 sin(i/j) cos(ij)/j and bi = n sin(i). The results are given in Table 2.
In the third series, we took the elements aij = 1/(i + j) + 2 sin(i/j) cos(ij)/j
and bi = n sin(i) as above, but also chose
N

h(x) = |xi |.
i=1
So, the cost function is non-smooth. Here we took versions with the Armijo
line-search. The results are given in Table 3, where (cl) now denotes the total
number of calculations of partial derivatives of fl . Therefore, (DNS) showed
rather stable and rapid convergence, and the explicit preference over (GNS) if
the dimensionality was greater than 20.
178 I. Konnov
Table 1. The numbers of iterations (it) and partial derivatives calculations (cl)
(GNS) (DNS)
it cl it cl
N =2 2 4 2 7
N =5 10 50 19 64
N = 10 17 170 67 235
N = 20 36 720 194 688
N = 40 105 4200 734 2935
N = 80 228 18240 3500 11241
N = 100 201 20100 4641 16473
(GNS) (DNS)
it cl it cl
N =2 6 12 2 7
N =5 10 50 12 49
N = 10 19 190 25 139
N = 20 38 760 55 349
N = 40 72 2880 119 871
N = 80 164 13120 309 2461
N = 100 252 25200 414 3279
(GNS) (DNS)
it cl it cl
N =2 13 26 7 23
N =5 16 80 32 84
N = 10 16 160 76 245
N = 20 40 800 257 982
N = 40 72 2880 485 1923
N = 80 135 10800 1127 4286
N = 100 188 18800 1374 6075
6 Conclusions
We described a new class of coordinate-wise descent splitting methods for limit
decomposable composite optimization problems involving set-valued mappings
and non-smooth functions. The method is based on selective coordinate variations
together with changing the tolerance parameters corresponding to a sequence of

mixed variational inequality problems without explicit evaluation of accuracy of
approximate solutions. We proved convergence without special concordance rules
for all the parameters and tolerances. Series of computational tests conrmed
rather satisfactory convergence.
Acknowledgement. The results of this work were obtained within the state assign-
ment of the Ministry of Science and Education of Russia, project No. 1.460.2016/1.4.
In this work, the author was also supported by Russian Foundation for Basic Research,
project No. 16-01-00109 and by grant No. 297689 from Academy of Finland.
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RAMBO: Resource-Aware Model-Based
Optimization with Scheduling for Heterogeneous
Runtimes and a Comparison with Asynchronous
Model-Based Optimization
Helena Kotthaus1(B) , Jakob Richter2 , Andreas Lang1 , Janek Thomas3 ,

Bernd Bischl3 , Peter Marwedel1 , Jorg Rahnenfuhrer2 , and Michel Lang2
1
Department of Computer Science 12, TU Dortmund University,
Dortmund, Germany
helena.kotthaus@tu-dortmund.de
2
Department of Statistics, TU Dortmund University, Dortmund, Germany
3
Department of Statistics, LMU Munchen, Munich, Germany
Abstract. Sequential model-based optimization is a popular technique

for global optimization of expensive black-box functions. It uses a regres-
sion model to approximate the objective function and iteratively pro-
poses new interesting points. Deviating from the original formulation, it
is often indispensable to apply parallelization to speed up the computa-
tion. This is usually achieved by evaluating as many points per iteration
as there are workers available. However, if runtimes of the objective func-
tion are heterogeneous, resources might be wasted by idle workers. Our
new knapsack-based scheduling approach aims at increasing the eec-
tiveness of parallel optimization by ecient resource utilization. Derived
from an extra regression model we use runtime predictions of point eval-
uations to eciently map evaluations to workers and reduce idling. We
compare our approach to ve established parallelization strategies on
a set of continuous functions with heterogeneous runtimes. Our bench-
mark covers comparisons of synchronous and asynchronous model-based
approaches and investigates the scalability.
Keywords: Black-box optimization Model-based optimization

Global optimization Resource-aware scheduling Performance man-
agement Parallelization
1 Introduction
Ecient global optimization of expensive black-box functions is of interest to
many elds of research. In the engineering industry, computationally expensive
models have to be optimized; for machine learning hyperparameters have to
be tuned; and for computer experiments in general, expensive algorithms have
parameters that have to be optimized to obtain a well-performing algorithm
conguration. The problems of global optimization can usually be modeled by a
https://doi.org/10.1007/978-3-319-69404-7_13
RAMBO: Resource-Aware Model-Based Optimization 181
real-valued objective function f with a d-dimensional domain space. The chal-

lenge is to nd the best point possible within a very limited time budget.
Together with [1,22,23], Model-based optimization (MBO) [15] is a state-of-
the-art algorithm for expensive black-box functions. Starting on an initial design
of already evaluated congurations, a regression model guides the search to new
congurations by predicting the outcome of the black-box on yet unseen con-
gurations. Based on this prediction an inll criterion (also called acquisition
function) proposes a new promising conguration for evaluation. In each iter-
ation the regression model is updated on the evaluated congurations of the
previous iteration until the budget is exhausted. Jones et al. [15] proposed this
now popular Ecient Global Optimization (EGO) algorithm. EGO sequentially
adds points to the design using Kriging as a surrogate and the Expected Improve-
ment (EI) as an inll criterion. Following, other inll criteria [14], specializations
e.g. for categorical search spaces like in SMAC [10] and noisy optimization [20]
have been introduced.
For computer experiments, parallelization has become of increasing interest
to reduce the overall computation time. Originally, the EGO algorithm iter-
atively proposes one point to be evaluated after another. To allow for paral-
lelization, inll criteria and techniques (constant liar, Kriging believer, qEI [9],
qLCB [11], MOI-MBO [4]) have been suggested that propose multiple points
in each iteration. Usually, the number of proposed points equals the number of
available CPUs. However, these methods still do use the available resources inef-
ciently if the runtime of the black-box is heterogeneous. Before new proposals
can be generated, the results of all evaluations within one iteration are gathered
to update the model. Consequently all CPUs have to wait for the slowest func-
tion evaluation before receiving a new point proposal. This can lead to idling
CPUs that are not contributing to the optimization. The goal in general is to
use all available CPU time to solve the optimization problem.
One approach to avoid idling is to desynchronize the model update. Here,
the model is updated each time an evaluation has nished, letting each parallel
worker propose the next point for evaluation itself. Such asynchronous techniques
have been suggested and discussed by [8,12]. The main challenge is to modify the
inll criterion to deal with points that are currently under evaluation to avoid
evaluations of very similar congurations. The Expected Expected Improvement
(EEI) [13] is one possibility for such a modication.
Another strategy is to keep the synchronous model update and schedule the
evaluations of the proposed points in such a way that idling is reduced. Such an
approach was presented in [19]. Here, a second regression model is used to predict
runtimes for the proposed points which are used as an input for scheduling.
Our article contains the following contributions: First, we extended the par-
allel, resource-aware synchronous model-based optimization strategy proposed
in [19] with an improved resource-aware scheduling algorithm. This algorithm,
which replaces the original simple rst t heuristic, is based on a knapsack solver
to better handle heterogeneous runtimes. Furthermore we use a clustering-based
renement strategy to ensure improved spatial diversity of the evaluated points.
182 H. Kotthaus et al.
Second, we compare our algorithm to three asynchronous MBO strategies

that also aim at using all available CPU time to solve the optimization problem
in parallel. Two of them [8,12] use Kriging as a surrogate and the third is included
in SMAC [10] which uses a random forest surrogate.
Third, we benchmark the MBO algorithms on a set of established continuous
test functions combined with simulated runtimes. For each function we use a
2- and a 5-dimensional version each of which is optimized using 4 and 16 CPUs
in parallel to investigate scalability.
Compared to the considered asynchronous approaches, our new approach
converges faster to the optima if the runtime estimates used as input for schedul-
ing are reliable.
2 Model-Based Global Optimization
The aim of global optimization is to nd the global minimum of a given function

f : X R, f (x) = y, x = (x1 , . . . , xd )T . Here, we assume X Rd , usually
expressed by simple box constraints. The optimization is guided by a surrogate
model which estimates the response surface of the black-box function f (see
also [22,23]). The surrogate is comparably inexpensive to evaluate and is utilized
to propose new promising points x in an iterative fashion. A promising point
x is determined by optimizing some inll criterion. After, f (x ) is evaluated to
obtain the corresponding objective value y, the surrogate model is retted and a
new point is proposed. The inll criterion quanties the potential improvement
based, on an exploitation-exploration trade-o where a low (good) expected
value of the solution (x) is rewarded, and low estimated uncertainty s(x) is
penalized. A popular inll criterion, especially for Kriging surrogate models, is
the expected improvement
EI(x) = E(max(ymin (x), 0))

ymin (x)) ymin (x)
= (ymin (x)) + s(x) ,
s(x) s(x)
where is the distribution and is the density function of the standard normal
distribution and ymin is the best observed function value so far. Alternatively,
the comparably simpler lower condence bound criterion
LCB(x, ) = (x) s(x), R
is used, where (x) denotes the posterior mean and s(x) the posterior standard
deviation of the regression model at point x. Before entering the iterative process,
initially some points have to be pre-evaluated. These points are generally chosen
in a space-lling manner to uniformly cover the input space. The optimization
usually stops after a target objective value is reached or a predened budget is
exhausted [22,23].
2.1 Parallel MBO

Ordinary MBO is sequential by design. However, applications like hyperparame-
ter optimization for machine learning or computer simulations have driven the
rapid development of extensions for parallel execution of multiple point eval-
uations. The parallel extensions either focus on a synchronous model update
using inll criteria with multi-point proposals or implement an asynchronous
evaluation where each worker generates one new point proposal individually.
Multi-Point proposals derive not only one single point x from a surrogate model,
but q points x1 , . . . , xq simultaneously. The q proposed points must be su-
ciently dierent from each other to avoid multiple evaluations with the same
conguration. For this reason Hutter et al. [11] introduced the criterion
qLCB(x, j ) = (x) j s(x) with j Exp() (1)
as an intuitive extension of the LCB criterion using an exponentially distributed

random variable. Since guides the trade-o between exploration and exploita-
tion, sampling multiple dierent j might result in dierent best points by
varying the impact of the standard deviation. The qLCB criterion was imple-
mented in a distributed version of SMAC [11]. An extension of the EI criterion
is the qEI criterion [9] which directly optimizes the expected improvement over
q points. A closed form solution to calculate qEI exists for q = 2 and useful
approximations can be applied for q 10 [7]. However, as the computation is
using Monte Carlo sampling, it is quite expensive. A less expensive and popu-
lar alternative is Kriging believer approach [9]. Here, the rst point is proposed
using the single-point EI criterion. Its posterior mean value is treated as a real
value of f to ret the surrogate, eectively penalizing the surrounding region
with a lower standard deviation for the next point proposal using EI again. This
is repeated until q proposals are generated.
In combination with parallel synchronous execution the above described
multi-point inll approaches can lead to underutilized systems because a new
batch of points can only be proposed as soon as the slowest function evaluation
is terminated. Snoek et al. [21] introduce the EI per second to address hetero-
geneous runtimes. The runtime of a conguration is estimated using a second
surrogate model and a combined inll criterion can be constructed which favors
less expensive congurations.
We also use surrogate models to estimate resource requirements but instead of
adapting the inll criterion, we use them for the scheduling of parallel function
evaluations. Our goal is to guide MBO to interesting regions in a faster and
resource-aware way without directly favoring less expensive congurations.
Asynchronous Execution approaches the problem of parallelizing MBO from a

dierent angle. Instead of evaluating multiple points in batches to synchronously
ret the model, the model is retted after each function evaluation to increase
CPU utilization workers. Here, each worker propose the next point for evaluation
itself, even when congurations xbusy are currently under evaluation on other
processing units. The busy evaluations have to be taken into account by the
surrogate model to avoid that new point proposals are identical or very similar
to pending evaluations. Here, the Kriging believer approach [9] can be applied to
block these regions. Another theoretically well-founded way to impute pending
values is the expected EI (EEI) [8,13,21]. The unknown value of f (xbusy ) is inte-
grated out by calculating the expected value of ybusy via Monte Carlo sampling,
which is, similar to qEI, computationally demanding. For each Monte Carlo iter-
ation values y1,busy , . . . , y,busy are drawn from the posterior distribution of the
surrogate regression model at x1,busy , . . . , x,busy , with denoting the number
of pending evaluations. These values are combined with the set of already known
evaluations and used to t the surrogate model. The EEI can then simply be
calculated by averaging the individual expected improvement values that are
formed by each Monte Carlo sample:
n
sim
= 1
EEI(x) EIi (x) (2)
nsim i=1
whereas nsim denotes the number of Monte Carlo iterations.

Besides the advantage of an increased CPU utilization, asynchronous execu-
tion can also potentially cause additional runtime overhead due to the higher
number of model rets, especially when the number of workers increases. There-
fore our experiments include a comparison with most of the above described
approaches to investigate the advantages and disadvantages.
Instead of using asynchronous execution to eciently utilize parallel com-
puter architectures, our new approach uses the synchronous execution combined
with resource-aware scheduling and is presented in the following section.
3 Resource-Aware Scheduling with Synchronous Model

Update
The goal of our new scheduling strategy is to guide MBO to interesting regions
in a faster and resource-aware way. To eciently map jobs (proposed points)
to available resources our strategy needs to know the resource demands of jobs
before execution. Therefore, we estimate the runtime of each job using a regres-
sion model. Additionally, we calculate an execution priority for each job based
on the multi-point inll criterion. In the following, we will describe these inputs.
3.1 Infill Criterion - Priority

The priorities of the proposed points should reect their usefulness for optimiza-
tion. In our setup we opt for the qLCB (1) to generate a set of job proposals
by optimizing the LCB for q randomly drawn values of j Exp( 12 ), as in
Richter et al. [19]. qLCB is suitable because the proposals are independent of
each other. There is no direct order of the set of obtained candidates xj in
terms of how promising or important one candidate is in comparison to each
other. Therefore, we introduce an order that steers the search more towards
promising areas. We give the highest priority to the point xj that was proposed
using the smallest value of j . We dene the priority for each point as pj := j .
3.2 Resource Estimation
To estimate the resource demands of proposed candidates, we use a separate

regression model. To adapt to the domain space of the objective function, we
choose the same regression method used for the surrogate. In the same fashion
as for the MBO algorithm, runtimes are predicted in each MBO iteration based
on all previously evaluated jobs and measured runtimes.
3.3 Resource-Aware Knapsack Scheduling
The goal of our scheduling strategy is to reduce the CPU idle time on the workers
while acquiring the feedback of the workers in the shortest possible time to avoid
model update delay. The set of points proposed by the multi-point inll criterion
forms the set of jobs J = {1, . . . , q} that we want to execute on the available
CPUs K = {1, . . . , m}. For each job the estimated runtime is given by tj and the
corresponding priority is given by pj . To reduce idle times caused by evaluations
of jobs with a low priority, the maximal runtime for each MBO iteration is dened
by the runtime of the job with the highest priority. Lower prioritized jobs have
to subordinate. At the same time we want to maximize the prot, given by the
priorities, of parallel job executions for each model update. To solve this problem,
we apply the 0 1 multiple knapsack algorithm by interfacing the R-package
adagio for global optimization routines [5]. Here the knapsacks are the available
CPUs and their capacity is the maximally allowed computing time, dened by
the runtime of the job with the highest priority. The items are the jobs J, their
weights are the estimated runtimes tj and their values are the priorities pj . The
capacity for each CPU is accordingly tj , with j := arg maxj pj . To select the
best subset of jobs the algorithm maximizes the prot Q:

Q= pj ckj ,
jJ kK
which is the sum of priorities of the selected jobs, under the restriction of the
capacity
tj tj ckj k K
jJ
per CPU. The restriction with the decision variable ckj {0, 1}

1 ckj j J, ckj {0, 1}.
kK
ensures that a job j is at most mapped to one CPU.

As the job with the highest priority denes the time bound tj it is mapped to
the rst CPU k = 1 exclusively and single jobs with higher runtimes are directly
discarded. Then the knapsack algorithm is applied to assign the remaining can-
didates in J to the remaining m1 CPUs. This leads to the best subset of J that
can be run in parallel minimizing the delay of the model update. If a CPU is left
without a job we query the surrogate model for a job with an estimated runtime
smaller or equal to tj to ll the gaps. For a useful scheduling the set of can-
didates should have considerably more candidates q than available CPUs. This
knapsack scheduling is a direct enhancement of the rst t scheduling strategy
presented in [19].
3.4 Refinement of Job Priorities via Clustering

The renement of job priorities has the goal to avoid parallel evaluations of
very similar congurations. Approaches to specically propose points that are
promising but yet diverse are described in [4]. qLCB performed well and was
chosen here because it is comparably inexpensive to create many independent
candidates. However, qLCB does not include a penalty for the proximity of
selected points which gets problematic if the number of parallel points is high.
Therefore, we use a distance measure to reprioritize pj to pj , encouraging the
selection sets of candidates more scattered in the domain space.
First, we oversample a set of q > m candidate points from the qLCB criterion
and partition them into q < q clusters using the Euclidean distance. Next, we
take the candidate with maximum priority pj from each cluster and sort them
according to their priority before pushing them to the list J of selected jobs.
Selected jobs are removed from the clusters and empty clusters are eliminated.
Finally,
We repeat this procedure until we have moved all q jobs into the list J.
we assign new priorities pj based on the order of J, i.e. the rst job in J gets

the highest priority q and the last job gets the lowest priority 1.
As a result, the set of candidates contains batches of jobs with similar pri-
ority that are spread in the domain space. The new priorities serve as input for
scheduling which groups the q jobs to m CPUs using the runtime estimates t.
4 Numerical Experiments
In our experiments, we consider two categories of synthetic functions to ensure
a fair comparison in a disturbance-free environment. They are implemented in
the R package smoof [6]:
1. Functions with a smooth surface: rosenbrock(d) and bohachevsky(d) with
dimension d = 2, 5. They are likely to be tted well by the surrogate.
2. Highly multimodal functions: ackley(d) and rastrigin(d) (d = 2, 5). We
expect that surrogate models can have problems to achieve a good t here.
As these are illustrative test functions, they have no signicant runtime. As
a resort, we also use these functions to simulate runtime behavior. First, we
combine two functions: One determines the number of seconds it would take
to calculate the objective value of the other function. E.g., for the combina-
tion rastrigin(2).rosenbrock(2) it would require rosenbrock(2)(x) seconds
to retrieve the objective value rastrigin(2)(x) for an arbitrary proposed point
x. Technically, we just sleep rosenbrock(2)(x) seconds before returning the
objective. We simulate the runtime with either rosenbrock(d) or rastrigin(d)
and analyze all combinations of our four objective functions, except where the
objective and the time function are identical.
A prerequisite for this approach is the unication of the input space. Thus,
we simply mapped values from the input space of the objective function to the
input space of the time function. The output of the time functions is scaled to
return values between 5 min to 60 min.
We examine the capability of the considered optimization strategies to mini-
mize functions with highly heterogeneous runtimes within a limited time budget.
To do this, we measure the distance between the best found point at time t and
a predened target value. We call this measure accuracy. In order to make this
measure comparable across dierent objective functions, we scale the function
values to [0, 1] with zero being the target value. It is dened as the best y reached
by any optimization method after the complete time budget. The upper bound
1 is the best y found in the initial design (excluding the initial runs of smac)
which is identical for all algorithms per given problem. Both values are averaged
over the 10 replications.
If an algorithm needs 2 h to reach an accuracy of 0.5, this means that within
2 h half of the way to 0 has been accomplished, after starting at 1. We compare
the dierences between optimizers at the three accuracy levels 0.5, 0.1 and 0.01.
The optimizations are repeated 10 times and conducted on m = 4 and m = 16
CPUs. We allow each optimization to run for 4 h on 4 CPUs and for 2 h on
16 CPUs in total which includes all computational overhead and idling. All
computations were performed on a Docker Swarm cluster using the R package
batchtools [18]. The initial design is generated by Latin hypercube sampling
with n = 4 d points and all of the following optimizers start with the same
design in the respective repetition:
rs: Random search, serving as base-line.
qLCB: Synchronous approach using qLCB where in each iteration q = m
points are proposed.
ei.bel: Synchronous approach using Kriging believer where in each iter-
ation m points are proposed.
asyn.eei: Asynchronous approach using EEI (100 Monte Carlo iterations)
asyn.ei.bel: Asynchronous Kriging believer approach.
rambo: Synchronous approach using qLCB with our new scheduling app-
roach where in each iteration q = 8 m candidates are proposed.
qLCB and ei.bel are implemented in the R package mlrMBO [3], which builds
upon the machine learning framework mlr [2]. asyn.eei, asyn.ei.bel and
rambo are also based on mlrMBO. We use a Kriging model from the package
DiceKriging [20] with a Matern 52 -kernel for all approaches above and add a
nugget eect of 108 Var(y), where y denotes the vector of all observed func-
tion outcomes. Additionally we compare our implementations to:
smac: Asynchronous approach that turns m independent SMAC runs into m

dependent runs by sharing surrogate model data (also called shared-
model-mode1 ).
SMAC was allowed the same initial budget as the other optimizers and was
started with the defaults and the shared-model-mode activated. SMAC uses a
random forest as surrogate and the EI criterion.
4.1 Quality of Resource Estimation
The quality of resource-aware scheduling naturally depends on the accuracy

of the resource estimation. Without reliable runtime predictions, the sched-
uler is unable to optimize for ecient utilization. As Fig. 1 exemplary shows,
the runtime prediction for the rosenbrock(5) time function works well as the
residual values are getting smaller over time, while the runtime prediction for
rastrigin(5) is comparably imprecise. For the 2-dimensional versions the results
are similar. This encourages to consider scenarios separately where runtime pre-
diction is possible (rosenbrock(), Subsect. 4.2) and settings where runtime pre-
diction is error-prone (rastrigin(), Subsect. 4.3) for further analysis.
bohachevsky.rastrigin_5d bohachevsky.rosenbrock_5d
prediction runtime (seconds)
1500
1000
500
500
1000
0 1 2 3 4 0 1 2 3 4
hours
Fig. 1. Residuals of the runtime prediction in the course of time for the rosenbrock(5)
and rastrigin(5) time functions on 4 CPUs and bohachevsky(5) as objective function.
Positive values indicate an overestimated runtime and negative values an underestima-
tion.
4.2 High Runtime Estimation Quality: rosenbrock
Figure 2 shows boxplots for the time required to reach the three dierent accuracy
levels in 10 repetitions within a budget of 4 h real time on 4 CPUs (upper part) and
1
Hutter, F., Ramage, S.: Manual for SMAC version v2.10.03-master. Department
of Computer Science, UBC. (2015), www.cs.ubc.ca/labs/beta/Projects/SMAC/
v2.10.03/manual.pdf.
Table 1. Ranking for accuracy levels 0.5, 0.1, 0.01 averaged over all problems with
rosenbrock() time function on 4 and 16 CPUs with a time budget of 4 h and 2 h,
respectively.
Algorithm 4 CPUs 16 CPUs

0.5 0.1 0.01 0.5 0.1 0.01
asyn.eei 3.32 (2) 3.52 (1) 4.97 (2) 3.75 (3) 4.30 (3) 5.45 (3)
asyn.ei.bel 3.55 (3) 4.10 (3) 4.97 (2) 3.48 (2) 4.08 (2) 4.53 (2)
RAMBO 3.17 (1) 3.85 (2) 4.57 (1) 3.13 (1) 3.93 (1) 4.47 (1)
ei.bel 4.38 (4) 4.98 (4) 5.90 (5) 5.00 (5) 5.48 (6) 6.28 (6)
qLCB 4.52 (5) 5.03 (5) 5.63 (4) 4.72 (4) 5.17 (4) 6.10 (4)
rs 6.02 (6) 6.67 (6) 6.83 (7) 5.50 (7) 6.48 (7) 6.87 (7)
smac 6.22 (7) 6.70 (7) 6.82 (6) 5.32 (6) 5.47 (5) 6.17 (5)
2 h on 16 CPUs (lower part). The faster an optimizer reaches the desired accuracy
level, the lower the displayed box and the better the approach. If an algorithm did
not reach an accuracy level within the time budget, we impute with the respective
time budget (4 h or 2 h) plus a penalty of 1000 s.
Table 1 lists the aggregated ranks over all objective functions, grouped by
algorithm, accuracy level, and number of CPUs. For this computation, the algo-
rithms are ranked w.r.t. their performance for each replication and problem
before they are aggregated with the mean. If there are ties (e.g. if an accuracy
level was not reached), all values obtain the worst possible rank.
The benchmarks indicate an overall advantage of our proposed resource-
aware MBO algorithm (rambo): On average, rambo reaches the accuracy level
rst in 2 of 3 setups on 4 CPUs and is always fastest on 16 CPUs. rambo is
closely followed by the asynchronous variant asyn.eei on 4 CPUs but the lead
becomes more clear on 16 CPUs. In comparison to the conventional synchronous
algorithms (ei.bel, qLCB), rambo as well as asyn.eei and asyn.ei.bel reach
the given accuracy levels in shorter time. This is especially true for objective
functions that are hard to model (ackley(), rastrigin()) by the surrogate
as seen in Fig. 2. The simpler asyn.ei.bel performs better than asyn.eei on
16 CPUs. Except for smac, all presented MBO methods outperform base-line rs
on almost all problems and accuracy levels. The bad average results for smac
are partly due to its low performance on the 5d problems and probably because
of the disadvantage of using a random forest as a surrogate on purely numerical
problems. A recent benchmark in [3] was able to demonstrate the competitive
performance of the Kriging based EGO approach. On 16 CPUs smac performs
better than rs and comparable to ei.bel.
For a thorough analysis of the optimization, Fig. 3 exemplary visualizes the
mapping of the parallel point evaluations (jobs) for all MBO algorithms on
16 CPUs for the 5d versions of the problems. Each gray box represents com-
putation of a job on the respective CPU. For the synchronous MBO algorithms
Algorithm asyn.eei asyn.ei.bel RAMBO ei.bel qLCB rs smac
ackley.rosenbrock bohachevsky.rosenbrock rastrigin.rosenbrock

4
2d (4 CPUs)
3
2
1
hours
0
4
5d (4 CPUs)
3
2
1
0
0.5 0.1 0.01 0.5 0.1 0.01 0.5 0.1 0.01
ackley.rosenbrock bohachevsky.rosenbrock rastrigin.rosenbrock
2.0
2d (16 CPUs)
1.5
1.0
0.5
hours
0.0
2.0
5d (16 CPUs)
1.5
1.0
0.5
0.0
0.5 0.1 0.01 0.5 0.1 0.01 0.5 0.1 0.01
accuracy level
Fig. 2. Accuracy level vs. execution time for dierent objective functions using time
function rosenbrock() (lower is better).
(rambo, qLCB, ei.bel) the vertical lines indicate the end of an MBO iteration.
For asyn.eei red boxes indicate that the CPU is occupied with the point pro-
posal. The necessity of a resource estimation for jobs with heterogeneous run-
times becomes obvious, as qLCB and ei.bel can cause long idle times by queuing
jobs together with large runtime dierences. The knapsack scheduling in rambo
manages to clearly reduce this idle time. This eect of ecient resource utiliza-
tion increases with the number of CPUs. rambo reaches nearly the same eective
resource-utilization as the asynchronous asyn.ei.bel algorithm and smac (see
Fig. 3) and at the same time reaches the accuracy level fastest on 16 CPUs.
The Monte Carlo approach asyn.eei generates a high computational over-
head as indicated by the red boxes, which reduces the eective number of evalu-
ations. Idling occurs because the calculation of the EEI is encouraged to wait for
ongoing EEI calculations to include their proposals. This overhead additionally
increases with the number of already evaluated points. asyn.ei.bel and smac
have a comparably low overhead and thus basically no idle time. This seems
to be an advantage for asyn.ei.bel on 16 CPUs where it performs better on
average than its complex counterpart asyn.eei.
ackley.rosenbrock_5d bohachevsky.rosenbrock_5d rastrigin.rosenbrock_5d

16
asyn.eei
12
8
4
0
16
asyn.ei.bel
12
8
4
0
16
RAMBO
12
8
4
CPU
0
16
12
ei.bel
8
4
0
16
12
qLCB
8
4
0
16
12
smac
8
4
0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
hours
Fig. 3. Scheduling of MBO algorithms. Time on x-axis and mapping of candidates to

m = 16 CPUs on y-axis. Each gray box represents a job. Each red box represents
overhead for the asynchronous approaches. The gaps represent CPU idle time. (Color
gure online)
Summed up, if the resource estimation that is used in rambo has a high
quality, rambo clearly outperforms the considered synchronous MBO algorithms
qLCB, ei.bel, and smac. This indicates, that the resource utilization obtained
by the scheduling in rambo leads to faster and better results, especially, when
the number of available CPUs increases. On average rambo performs better than
all considered asynchronous approaches.
4.3 Low Runtime Estimation Quality: rastrigin
The time function rastrigin used in the following scenario is dicult to t by

surrogate models, as visualized by the residual plot in Fig. 1. For this reason, the
benet of our resource-aware knapsack strategy is expected to be minimal. For
example in a possible worst case multiple supposedly short jobs are assigned to
one CPU but their real runtime is considerably longer.
Similar to Subsect. 4.2, Fig. 4 shows boxplots for the benchmark results, but
with rastrigin() as the time function. Table 2 provides the mean ranks for
Fig. 4, calculated in the same way as in previous Subsect. 4.2.
Despite possible wrong scheduling decisions, rambo still manages to outper-
form qLCB and performs better than ei.bel on the highest accuracy level on
average. asyn.eei reaches the accuracy levels fastest on 4 CPUs. Similar to the
previous benchmarks on Subsect. 4.2, the simplied asyn.ei.bel seems to ben-
et from its reduced overhead and places rst on 16 CPUs. This dierence w.r.t.
the scalability becomes especially visible on rosenbrock().
Table 2. Ranking for accuracy levels 0.5, 0.1, 0.01 averaged over all problems with
rastrigin() time function on 4 and 16 CPUs with a time budget of 4 h and 2 h,
respectively.
Algorithm 4 CPUs 16 CPUs

0.5 0.1 0.01 0.5 0.1 0.01
asyn.eei 3.65 (1) 3.25 (1) 4.47 (1) 4.42 (3) 4.38 (2) 5.20 (3)
asyn.ei.bel 3.88 (2) 3.50 (2) 4.52 (2) 3.90 (1) 3.75 (1) 4.33 (1)
RAMBO 4.50 (4) 4.70 (4) 4.72 (3) 4.43 (4) 4.60 (4) 5.17 (2)
ei.bel 4.22 (3) 4.42 (3) 4.87 (4) 4.33 (2) 4.55 (3) 5.27 (4)
qLCB 4.95 (5) 4.80 (5) 5.38 (5) 5.10 (5) 5.00 (5) 5.82 (5)
rs 6.30 (7) 6.42 (6) 6.63 (6) 5.78 (7) 6.23 (7) 6.43 (6)
smac 5.90 (6) 6.98 (7) 7.00 (7) 5.30 (6) 5.77 (6) 6.72 (7)
Algorithm asyn.eei asyn.ei.bel RAMBO ei.bel qLCB rs smac
ackley.rastrigin bohachevsky.rastrigin rosenbrock.rastrigin

4
2d (4 CPUs)
3
2
1
hours
0
4
5d (4 CPUs)
3
2
1
0
0.5 0.1 0.01 0.5 0.1 0.01 0.5 0.1 0.01
ackley.rastrigin bohachevsky.rastrigin rosenbrock.rastrigin
2.0
2d (16 CPUs)
1.5
1.0
0.5
hours
0.0
2.0
5d (16 CPUs)
1.5
1.0
0.5
0.0
0.5 0.1 0.01 0.5 0.1 0.01 0.5 0.1 0.01
accuracy level
Fig. 4. Accuracy level vs. execution time for dierent objective functions using time
function rastrigin() (lower is better).
smac can not compete with the Kriging based optimizers. Overall, rambo
appears not to be able to outperform the asynchronous MBO methods on 4 CPUs
as unreliable runtime estimates likely lead to suboptimal scheduling decisions.
However, rambo reaches comparable results to asyn.eei on 16 CPUs and com-

pared to the default synchronous approaches it is a viable choice.
5 Conclusion
We benchmarked our knapsack based resource-aware parallel MBO algorithm
rambo against popular synchronous and asynchronous MBO approaches on a set
of illustrative test functions for global optimization methods. Our new approach
was able to outperform SMAC and the default synchronous MBO approach qLCB
on the continuous benchmark functions. On setups with high runtime estimation
quality it converged faster to the optima than the competing MBO algorithms on
average. This indicates, that the resource utilization obtained by our new app-
roach improves MBO, especially, when the number of available CPUs increases.
On setups with low runtime estimation quality the asynchronous Kriging based
approaches performed best on 4 CPUs and only the simplied asynchronous
Kriging believer kept its lead on 16 CPUs. Unreliable estimates likely lead to sub-
optimal scheduling decisions for rambo. While the asynchronous Kriging believer
approach, SMAC and rambo beneted from increasing the number of CPUs, the
overhead of the asynchronous approach based on EEI increased.
If the runtime of point proposals is predictable we suggest our new rambo
approach for parallel MBO with high numbers of available CPUs. Even if the
runtime estimation quality is obviously hard to determine in advance, for real
applications like hyperparameter optimization for machine learning methods pre-
dictable runtimes can be assumed. Our results also suggest that on some setups
the choice of the inll criterion determines which parallelization strategy can
reach a better performance. For future work a criterion that assigns an inll
value to a set of candidates that can be scheduled without causing long idle
times appears promising. Furthermore we want to include the memory con-
sumption measured by the traceR [16,17] tool into our scheduling decisions for
experiments with high memory demands.
Acknowledgments. J. Richter and H. Kotthaus These authors contributed

equally. This work was partly supported by Deutsche Forschungsgemeinschaft (DFG)
within the Collaborative Research Center SFB 876, A3 and by Competence Network
for Technical, Scientic High Performance Computing in Bavaria (KONWIHR) in the
project Implementierung und Evaluation eines Verfahrens zur automatischen, massiv-
parallelen Modellselektion im Maschinellen Lernen.
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A New Constructive Heuristic for the No-Wait
Flowshop Scheduling Problem
Lucien Mousin(B) , Marie-Eleonore Kessaci, and Clarisse Dhaenens
Univ. Lille, CNRS, Centrale Lille, UMR 9189 - CRIStAL - Centre de Recherche en
Informatique Signal et Automatique de Lille, 59000 Lille, France
lucien.mousin@ed.univ-lille1.fr,
{me.kessaci,clarisse.dhaenens}@univ-lille1.fr
Abstract. Constructive heuristics have a great interest as they manage

to nd in a very short time, solutions of relatively good quality. Such
solutions may be used as initial solutions for metaheuristics for example.
In this work, we propose a new ecient constructive heuristic for the No-
Wait Flowshop Scheduling Problem. This proposed heuristic is based on
observations on the structure of best solutions of small instances as well
as on analyzes of ecient constructive heuristics principles of the liter-
ature. Experiments have been conducted and results show the eciency
of the proposed heuristic compared to ones from the literature.
1 Introduction
Scheduling problems is an important class of combinatorial optimization prob-

lems, most of them being N P hard. They consist in the allocation of dierent
operations on a set of machines over the time. The aim of scheduling problems is
to optimize dierent criteria such as the makespan or the owtime. Among these
scheduling problems, some of them, for example the jobshop or the owshop,
have been widely studied in the literature, as they represent many industrial
situations.
In this paper, we are specically interested in the No-Wait Flowshop Schedul-
ing Problem (NWFSP). This extension of the classical permutation Flowshop
Scheduling Problem (FSP) imposes that operations have to be processed without
any interruption between consecutive machines. This additional constraint aims
at describing real process constraints that may be found in the chemical indus-
try for example. Regarding the solving of the problem, this additional constraint
also introduces interesting characteristics that we propose to exploit within the
design of an ecient constructive heuristic.
Indeed, while solving N P hard problems, the use of exact methods, mostly
based on enumerations, is not practicable. Therefore heuristics and metaheuris-
tics are developed. Heuristics designed for a specic problem can use, for exam-
ple, priority rules to construct the schedule. Their advantages are their speed
most of them are greedy heuristics and their specicity characteristics of the
problem to be solved can be exploited. Metaheuristics, on their side, are generic
https://doi.org/10.1007/978-3-319-69404-7_14
A New Constructive Heuristic for the NWFSP 197
methods that can be applied to many optimization problems. Their eciency

is linked to their high ability of exploring the search space and the way the
metaheuristic has been adapted to the problem under study. The initial solu-
tion, when one is required, may also inuence the quality of the nal solution
provided.
Hence, as explained, the quality of solutions provided by heuristics and meta-
heuristics depends on the way these methods can integrate specicities of the
problem under study. Several approaches exist to integrate such specicities.
For example, it is possible to analyze o line characteristics of instances and
to identify the best heuristic (or parameters of metaheuristic) for each kind of
instances. Then, during the search, the method can identify the type of instance
to solve and adapt the heuristic to use. This has been successfully used, for
example, for a set partitioning problem [8]. Another approach deals with the
integration of knowledge about interesting structures of solutions within the
optimization method. In this context, the aim of the present work is to develop
such a heuristic method for the No-Wait Flowshop Scheduling Problem. Thus,
the contributions are the following: 1. analysis of interesting structure of optimal
solutions, 2. integration of these observations in the design of a new constructive
heuristic, 3. study the interest of the proposed approach when used alone or as
the initialization of a metaheuristic.
The remainder of this paper is organized as follows. Section 2 introduces the
problem formulation and provides a literature review on constructive heuristics
for the NWFSP. Section 3 presents in details the proposed constructive heuristic.
In Sect. 4 experiments are conducted and a comparison with ecient heuristics
of the literature is provided. Some conclusions and perspectives for future works
are given in Sect. 5.
2 The No-Wait Flowshop Scheduling Problem

2.1 Description of the Problem
The No-Wait Flowshop Scheduling Problem (NWFSP) is a variant of the well-

known Permutation Flowshop Scheduling Problem (FSP), where no waiting time
is allowed between the executions of a job on the successive machines. It may be
dened as follows: Let J be a set of n jobs that have to be processed on a set of
M ordered machines; pi,j is dened as the processing time of job i on machine j.
A solution of the NWFSP is a sequence of jobs and so, is commonly represented
by a permutation = {1 , . . . , n } where 1 is the rst job scheduled and n
the last one. In this paper, the goal is to nd a sequence that minimizes the
makespan criterion (recorded as Cmax ) dened as the total completion time of
the schedule.
Regarding the complexity of the problem, it has been proved by Wismer
that the NWFSP can be viewed as an Asymmetric Traveling Salesman Problem
(ATSP ) [17]. In addition, Rock [13] proved that for m-machines (m 3), the
NWFSP is NP-hard while, the 2-machines case can be solved in O(n log n) [4].
198 L. Mousin et al.
However, the NWFSP possesses a characteristic not present in the classical

FSP. Indeed, the start-up interval between two dened consecutive jobs of the
sequence on the rst machine is constant and does not depend on their position
within the sequence. This interval, called the delay between two jobs, has been
dened by Bertolissi [2]. Let di,i be the delay of two jobs i and i , it is computed
as follows:

r r1

di,i = pi,1 + max pi,j pi ,j , 0
1rm
j=2 j=1
This allows to compute the completion time Ci () of the job i of sequence

directly from the delays of the preceding jobs as follows:
i
m

Ci () = dk1 ,k + pi ,j (1)
k=2 j=1
where i {2, ..., n}. Note that C1 () is the sum of the processing times on
the m machines of the rst scheduled job and then, is not concerned by the
delay. Therefore, the makespan (Cmax ()) of a sequence can be computed
from Eq. (1) with a complexity of O(n).
Experiments we will present, involve local search methods at dierent steps.
These methods need a neighborhood operator to move from a solution to another.
Such neighborhood operators are specic to the problem and more precisely to
the representation of a solution. In this work, we use a permutation representa-
tion and a neighborhood structure based on the insertion operator [14]. For a
sequence , it consists in inserting a job i at position k (i = k). Hence, jobs
between positions i and k are shifted. The size of this neighborhood, i.e., the
number of neighboring solutions, is (n 1)2 . Two sequences and are said to
be neighbors when they dier from exactly one insertion move. It is very inter-
esting to note that exploiting the characteristics of the NWFSP, the makespan
of can be directly computed from the makespan of with a complexity of
O(1) [11].
2.2 State-of-the-Art
Many heuristics and metaheuristics have been proposed to solve scheduling prob-
lems and in particular the No-Wait Flowshop variant. The literature review pro-
posed here, focuses on constructive heuristics as it is the scope of the article.
First, let us note that the well-known NEH (Nawaz, Enscore, Ham) heuristic,
proposed for the classical permutation FSP [10], has been successfully applied on
the No-Wait variant. Moreover, constructive heuristics have been designed specif-
ically for the NWFSP. We may cite BIH (Bianco et al. [3]), BH (Bertolissi [2])
and RAJ (Rajendran [12]) among others. These heuristics dene the order in
which jobs are considered, regardingone criterion (e.g. decreasing order of sum
of processing time of each job i m pi,m ), but this may be improved with
other jobs orders, even random. Indeed, in some contexts, it has been shown
that repeating a random construction of solutions, may be more ecient than
applying a constructive heuristic [1].
All these heuristics construct good quality solutions that can be further
improved by a neighborhood-based metaheuristic, for example. In this paper,
we focus on NEH and BIH as they provide high quality solutions and are con-
sidered as references for this problem. These two heuristics will be used later in
the article for comparison.
The principle of NEH heuristic is to iteratively build a sequence of jobs J.
First, the n jobs of J are sorted by decreasing sums of processing times. Then,
at each iteration, the rst remaining unscheduled job is inserted in the partial
sequence in order to minimize the partial makespan. Algorithm 1 presents NEH.
Algorithm 1. NEH
Data: Set J of n jobs, the sequence of jobs scheduled
= ;
Sort the set J in decreasing sums of processing times;
for k = 1 to n do
Insert job Jk in at the position, which minimizes the partial makespan.
This is a greedy heuristic; only n iterations are needed to build a sequence. At

each iteration, a job Jk is inserted. The makespan of the new partial sequence,
may be computed with a complexity of O(n) thanks to the delay, when job Jk
is inserted at the rst position (see Sect. 2.1). Partial sequences, in which Jk is
inserted at another position in the sequence, are neighbors of the previous one
so, their makespan may be computed with a complexity of O(1), according to the
property exposed at the end of Sect. 2.1. Thus the complexity of one iteration of
NEH is O(n). Therefore, the complexity of the whole execution of NEH is O(n2 ).
BIH heuristic is another constructive heuristic based on the best insertion
of a job in the partial sequence. The main dierence with NEH is that all the
unscheduled jobs are tested to nd the best partial makespan. Indeed, while
unscheduled jobs are remaining, for each one, all the possible positions to insert
it into the partial sequence are evaluated; the job and the position that minimize
the partial makespan are chosen. Algorithm 2 presents BIH.
If we consider the schedule of a job as one iteration, BIH builds a sequence
with n iterations only. However, at each iteration, each job is a candidate to be
inserted at one of all the possible positions to nd the one that minimizes the partial
makespan. For each job, nding the position that minimizes the partial makespan
has a maximal complexity of O(n) so, the complexity of one iteration of BIH is
O(n2 ). Therefore, the complexity of the whole execution of BIH is O(n3 ).
BIH has been proposed in the literature after NEH in order to construct
better quality solutions. However, the complexity of BIH is O(n3 ) whereas the
one of NEH is O(n2 ). Thus, NEH is much faster than BIH, and for large size
problems a compromise between computational time and quality is required.
Algorithm 2. BIH
Data: Set J of n jobs, the sequence of jobs scheduled
= ;
while J = do
Find job k J, which can be inserted in the sequence , such that the
partial makespan is minimized. Let h be the best insertion position of job k
in the sequence ;
Insert job k at position h in the sequence ;
Remove k from set J;
3 IBI: Iterated Best Insertion Heuristic
Following the constructions principles of previously exposed heuristics and after

analyzing the structure of optimal solutions, we propose the Iterated Best Inser-
tion (IBI) heuristic.
3.1 Analysis of Optimal Solutions Structure
Most of constructive heuristics are based on the best insertion principle. Indeed,
the principle of the heuristic is to increase to increase, at each iteration, the
problem size by one. It starts with a sub-problem of size one (only one job to
schedule), then increases the size of the problem to solve (two jobs to schedule,
three jobs. . . ) until the size of the initial problem is reached.
In order to understand the dynamic of such a construction, we analyzed the
structure of consecutive constructed sub-sequences and compare them to optimal
sequences. Obviously this can be done only for rst steps, as it is impossible to
enumerate all the sequences, and nd the optimal one, when the problem size
is too large. Our proposition is to extend observations realized on sub-problems
to the whole problem, in order to provide better solution. This approach can be
compared to the Streamlined Constraint Reasoning, where the solution space
is partitioned into sub-spaces whose structures are analyzed to better solve the
whole problem [5].
Figure 1 gives an example starting from a sub-problem of size 8, P8 , where 8
jobs are scheduled in the optimal order to minimize the makespan. Then, follow-
ing the constructive principle, job 9 is inserted at the position that minimizes the
makespan leading to the sub-problem P9 of size 9. This sequence corresponds
to the one given by NEH strategy. When comparing the obtained sequence with
the optimal solution of P9 , we can observe that they are very close. Indeed, only
two improving re-insertions (re-insertions of jobs 7 and 2) are needed to obtain
this optimal solution from the NEH solution.
Fig. 1. Example of the evolution of the structure of the optimal solution from a sub-
problem P8 of size 8 to a sub-problem P9 of size 9.
Experimental analysis on dierent instances have conrmed the following

observations:
at each step the structure of the obtained solution is not far from the structure
of the optimal solution for the sub-problem,
a few improving neighborhood applications may often lead to this optimal
solution.
3.2 Design of IBI

Following these observations, we propose a new constructive heuristic called Iter-
ated Best Insertion (IBI). At each iteration, two phases are successively achieved:
(i) the rst remaining unscheduled job is inserted in the partial sequence in order
to minimize the partial makespan (same strategy than NEH) and (ii), an iter-
ative improvement is performed on this partial solution to re-order jobs and to
expect to be closer to the optimal solution of the sub-problem (see Algorithm 3).
An iterative improvement is a method that moves, using a neighborhood oper-
ator, from a solution to one of its improving neighbors, which optimizes the
quality. It naturally stops when a local optimum (a solution with no improving
neighbors) is reached. Here, the neighborhood operator is based on the insertion
and then possible move are applied while a better makespan is found.
Algorithm 3. IBI
Data: Set J of n jobs, the sequence of jobs scheduled, criterion to sort J,
cycle number of iterations without iterative improvement.
= ;
Sort set J according to ;
for k = 1 to n do
Insert job Jk in at the position which minimizes the partial makespan.;
if k 0 [cycle] then
Perform an iterative improvement from .
Two parameters have been introduced in the proposed method. First, a

criterion to initially sort the set of jobs J. It will indicate in which order jobs
will be considered during the construction. Second, cycle the number of itera-
tions without applying the iterative improvement procedure. Indeed, even if the
experimental analysis shows that the sequence built in phase (i) is not far from
the optimum of the sub-problem, it is known that the exploration of the neigh-
borhood is more and more time-consuming and the optimum more and more
dicult to reach with the increase of the problem size. In order to control the
time-performance of IBI, the cycle parameter allows the iterative improvement
to be executed at a regular number of iterations only.
3.3 Experimental Analysis of Parameters
As the initial sort of IBI , may impact its performance as well as the cycle
parameter, the performance of IBI is evaluated under these two parameters. This
part, rst presents the experimental protocol used, and then compare several
versions of the IBI algorithm, using several initial sorts and nally analyses the
impact of the cycle parameter.
Experimental Protocol. The benchmark used to evaluate the performance

of the dierent variants of IBI is the Taillards instances [15], initially provided
for the owshop problem but also widely used in the literature for the NWFSP.
This benchmark proposes 120 instances, organized by 10 instances of 12 dierent
sizes. Our experiments are conducted on the 10 available instances of size N M
where the number of jobs N is ranging from 20 to 200 and the number of machines
M from 5 to 20 (total of 110 instances).
To compare the algorithms, the Relative Percentage Deviation (RPD) with
the best known solution is computed. For a quality Q to minimize, the RPD is
computed as follows:
Q(Solution) Q(Best known solution)

RP D = 100 (2)
Q(Best known solution)
Because of its Iterative Improvement phase, IBI is a stochastic method. Thus,
30 runs are required to allow making conclusions about results obtained. To
follow this recommendation, all variants of the algorithm are executed 30 times
per instance and the statistical Friedman test is performed on the RPD of all
executions to compare algorithms.
Each algorithm is implemented using C++ and the experiments were exe-
cuted on an Intel(R) Xeon(R) 3.5 GHz processor.
IBI: Initial Sort. Greedy heuristics, such as IBI, consider jobs in a given order.
For example, in NEH, jobs are ordered by the decreasing sum of processing times.
The specicity of the NWFSP gives other useful information as the GAP between
two jobs on each machine that represents the idle time of the machine between
the two jobs [9]. This measure does not depend on the schedule. Hence, for each
machine, the GAP between each pair of jobs can be computed independently
from the solving. The sum of GAPs for a pair of jobs represents the sum of
their GAP on all the machines. To obtain a single value per job, we propose to
dene the total GAP of one job as the sum of the sum of GAPs. Hence, a high
value of total GAP for a job indicates that the job has no good matching in the
schedule. On the other hand, a low total GAP indicates that the job ts well
with the others. In order to evaluate the impact of the initial sort and to dene
the most ecient one for the IBI heuristic, we experiment several initial sorts ,
based on (i) the total GAP or on (ii) the sum of processing times in decreasing
or increasing order for both.
Table 1. Average RPD obtained on each size of Taillards instances for dierent sorts:
No Sort, Decreasing Mean GAP (DecrMeanGAP), Increasing Mean GAP (IncrMean-
GAP), Decreasing sum of processing times (DecrPI) and the increasing sum of process-
ing times (IncrPI). RPD values in bold stand for algorithms outperforming the other
ones according to the statistical Friedman test.
Instance NoSort DecrMeanGAP IncrMeanGAP DecrPI IncrPI

20 5 1.79 2.32 2.57 2.13 1.38
20 10 2.04 1.72 1.67 1.30 1.77
20 20 1.44 1.26 1.04 1.12 1.15
50 5 3.74 3.65 4.01 3.23 3.04
50 10 2.78 3.17 3.25 2.74 2.57
50 20 2.53 2.41 2.47 2.80 2.50
100 5 4.68 4.52 4.67 4.13 4.12
100 10 3.45 3.36 3.53 3.24 3.27
100 20 3.05 2.82 2.81 2.79 2.89
200 10 4.32 4.19 4.22 3.81 3.87
200 20 3.24 3.26 3.23 3.03 2.99
Table 1 provides a detailed comparison of the dierent initial sorts on the 110
Taillards instances. This table shows that when a signicant dierence is observed
the best initial sort is to consider the sum of processing times. Ordering jobs by
increasing or decreasing order has no signicant inuence. Thus for the following,
we choose the increasing sum of processing times as the initial sort of IBI.
IBI: Cycle. The use of an iterative improvement at each iteration can be time
expensive. In order to minimize the execution time, we introduce a cycle. A cycle
is a sequence of x iterations without any iterative improvement.
In Table 2, we study the impact of the size of the cycle on both the quality
of solutions and the execution time.
These results show that the quality decreases with a large cycle size, but
is obviously faster. As the objective of such a constructive heuristic is to pro-
vide in a very reasonable time a solution as good as possible, and as the time
required here even for large instances stays reasonable, we propose not to use
Table 2. Average RPD and time (milliseconds) obtained on Taillards instances for
dierent sizes of cycle. RPD values in bold stand for algorithms outperforming the
other ones according to the Friedman test. A cycle of size x, indicates the iterative
improvement is executed every x iterations. Thus, a cycle of size n indicates the iterative
improvement is executed only once, at the end of the construction.
Instance 1 2 5 10 n
RPD Time RPD Time RPD Time RPD Time RPD Time
20 5 1.38 0.77 1.24 0.55 1.43 0.27 1.78 0.20 1.62 0.18
20 10 1.77 0.83 1.82 0.52 1.85 0.26 1.49 0.20 1.69 0.19
20 20 1.15 0.77 1.35 0.47 1.71 0.25 1.74 0.19 1.68 0.16
50 5 3.04 12.16 3.25 6.76 3.46 3.52 3.62 2.49 3.94 1.74
50 10 2.57 11.55 2.49 6.69 2.75 3.62 3.07 2.39 3.26 1.48
50 20 2.50 11.73 2.37 6.67 2.58 3.58 2.65 2.42 3.14 1.59
100 5 4.12 95.74 4.22 55.96 4.42 30.22 4.65 19.40 5.29 9.62
100 10 3.27 93.43 3.27 54.56 3.30 29.19 3.45 18.78 4.17 9.71
100 20 2.89 92.90 2.93 53.60 3.07 28.35 3.16 18.28 3.56 9.88
200 10 3.87 755.09 3.91 435.40 4.07 225.52 4.09 146.26 4.74 55.04
200 20 2.99 746.62 3.10 431.33 3.11 222.80 3.18 146.26 3.93 57.83
any cycle, that is to say to execute the iterative improvement at each step of the
construction.
In conclusion of these experiments we propose to x for the remainder of this
work as IBI parameters: an initial sort based on the increasing sum of processing
times and no cycle (i.e. cycle of size 1).
4 Experiments
The aim of this section is to analyze the eciency of the proposed IBI method in
two situations. First, IBI alone will be compared to other constructive heuristics
of the literature, and in a second time these dierent heuristics will be used as
initialization of a classical local search. Along this section, the same experimental
protocol as before is used, and parameters used for IBI are those resulting from
experiments of Sect. 3.3.
4.1 Eciency of IBI

We choose to compare IBI with two other heuristics of the literature: NEH and
BIH. Indeed, as exposed in Sect. 2.2 NEH and BIH are both interesting and
ecient constructive heuristics for the NWFSP:
1. NEH is a classical heuristic for ow-shop problems and it has the advantage
to be very fast. Moreover, the proposed method IBI can be viewed as an
improvement of NEH, as an iterative improvement is added at each step.
2. BIH is a classical heuristic for the NWFSP and very ecient as, according to
some preliminary experiments, it oers the best performance among several
tested heuristics.
Table 3 shows the comparative study between IBI and the two other construc-
tive heuristics. Both RPD and computation time are given to exhibit the gain in
quality with regards to the time. NEH and BIH are deterministic and so, both val-
ues of an instance size are average ones computed from the ten RPD values obtained
for the 10 instances respectively. On the other hand, IBI values correspond to the
average ones over the 30 runs and the 10 instances of each instance size.
Table 3. RPD and time (milliseconds) obtained on Taillards instances for NEH, BIH
and IBI (average values). RPD values in bold stand for algorithms statistically outper-
forming the other ones according to the Friedman test.
Instance NEH BIH IBI

RPD Time RPD Time RPD Time
20 5 3.95 0.03 3.03 0.10 1.38 0.77
20 10 4.18 0.02 3.60 0.09 1.77 0.82
20 20 2.92 0.02 1.74 0.08 1.15 0.76
50 5 6.84 0.10 5.98 1.10 3.04 12.16
50 10 4.59 0.09 4.10 1.13 2.57 11.54
50 20 4.84 0.10 4.01 1.18 2.50 11.72
100 5 8.14 0.32 6.85 9.07 4.12 95.74
100 10 6.10 0.33 5.66 9.29 3.27 93.43
100 20 5.17 0.33 4.65 9.30 2.89 92.90
200 10 6.72 1.21 5.85 72.44 3.87 755.09
200 20 5.74 1.21 4.51 71.95 2.99 746.62
Solutions built by IBI have a better quality (lower RPD) than those built by
NEH or BIH. It has statistically been veried with the Friedman Test. Beside,
IBI was able to better perform than NEH or BIH for 106 instances over the 110
available instances. The counterpart of this performance is the computational
time required to execute IBI regarding to the two other heuristics. However,
this time remains reasonable as, less than one second is required even for larger
instances. IBI is an ecient alternative to the classical heuristics used to build
good quality solutions.
4.2 IBI as Initialization of a Local Search
A general use of constructive heuristics is to execute them as an initialization

phase of meta-heuristics. In order to evaluate the pertinence of using IBI in such
an initialization phase, it has been combined with a Tabu Search (TS). This
local search is widely used to solve owshop problems [6,7,16] and so is a good
candidate to show the pertinence to build solutions with IBI rather than NEH
or BIH. The three heuristics are combined with TS. In order to be fair, every
combined approaches have the same running time xed to 1000 n ms (with n
the number of jobs). These experiments aim at checking if the quality reached
is enough to justify the use of IBI as initialization or if its higher execution time
penalizes its use.
Table 4 presents results about the average RPD values among the 30 runs
for each combined approach for each instance size. Results exposed in this table
compared to those of Table 3 rst indicate that the Tabu Search manages to
improve solutions produced by the dierent heuristics. It also indicates that IBI
used as an initialization phase, gets better results than the others, even if the
dierence is not always statistically proved. In particular for instances with a
high number of jobs to schedule, IBI shows a very good performance.
Table 4. Average RPD on Taillards instances for NEH+TS, BIH+TS and IBI+TS.
RPD values in bold stand for algorithms statistically outperforming the other ones
according to the Friedman test.
Instance NEH+TS BIH+TS IBI+TS

20 5 1.12 1.30 0.83
20 10 1.30 1.35 1.39
20 20 1.15 0.71 0.73
50 5 3.17 3.49 2.63
50 10 2.68 2.53 2.15
50 20 2.53 2.41 2.03
100 5 4.66 4.55 3.78
100 10 3.70 3.48 2.94
100 20 2.97 2.93 2.62
200 10 4.25 4.20 3.67
200 20 3.56 2.99 2.77
As an illustration, Fig. 2 shows the evolution of the average makespan for

a particular instance with 200 jobs and 20 machines (mean of the 30 runs).
Regarding only the quality after the initialization, IBI is better than the two
other heuristics, as shown in Table 3 previously. After the improvement of the
Tabu Search for the two other heuristics, the nal quality seems to be equivalent
to the one obtained by IBI without applying the local search. That shows the
eciency of this approach. Finally, the Tabu Search does not help a lot IBI.
Indeed, thanks to the successive local improvement in the second phase of IBI,
the solution provided by IBI is already a local optimum. Therefore, it is dicult
for the Tabu Search to improve the solution obtained with IBI.
Fig. 2. Evolution of the average makespan on 30 runs for the 7th instance of Taillard
(200 jobs, 20 machines) for the three combined approaches. IBI+TS is represented by
squares, NEH+TS by crosses and BIH+TS by circles.
In conclusion, these experiments show the good performance of IBI for both
aspects: as a constructive heuristics and as initialization of a meta-heuristic. In
particular, they show the contribution of the local improvement to build the
solution.
5 Conclusion and Perspectives
This work presents IBI, a new heuristic to minimize the makespan for the No-
Wait Flowshop Scheduling Problem. IBI has been designed from the analysis of
existing heuristics of the literature as well as of the structure of optimal solutions.
Indeed, IBI is an improvement of the widely used heuristic (NEH), where an
iterative improvement procedure has been added after each insertion of a job.
Even if this additional procedure increases the computational time compared to
other constructive heuristics of the literature (NEH and BIH), the improvement
of the quality of the nal solution built is noticeable and has statistically been
validated with the Friedman test.
Then, we analyzed if this extra computational time is justied in term of
quality obtained by evaluating the capacity of IBI to provide a good solution.
IBI and the others heuristics have been used as initialization for a metaheuristic
i.e., the initial solution of the approach is a solution built by a heuristic. The
experiments, on the Taillard instances, show that IBI helps the metaheuristic to
be more ecient and the extra time needed to build an initial solution is not a
drawback.
What is interesting in this work is that the addition of a simple modica-

tion of an existing constructive heuristic improves the results. This modication
has been proposed from properties observed which make the application of an
iterative improvement procedure at each step 1. useful only a few steps are
required to reach the optimal solutions for the small sub-problems and 2. no
time consumingthe makespan of a neighborhood solution can be computed in
O(1). We can now wonder whether such a modication can be performed to other
constructive heuristics for other variants of the owshop or more generally, for
other problems (if they also present equivalent properties).
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Sharp Penalty Mappings for Variational
Inequality Problems
Evgeni Nurminski(B)
School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia

nurminski.ea@dvfu.ru
Abstract. First, this paper introduces a notion of a sharp penalty map-

ping which can replace more common exact penalty function for convex
feasibility problems. Second, it uses it for solution of variational inequal-
ities with monotone operators or pseudo-varitional inequalities with ori-
ented operators. Appropriately scaled the sharp penalty mapping can be
used as an exact penalty in variational inequalities to turn them into
fixed point problems. Then they can be approximately solved by simple
iteration method.
Keywords: Monotone variational inequalities Oriented mappings

Sharp penalty mappings Exact penalty mappings Approximate solu-
tion
1 Introduction
Variational inequalities (VI) became one of the common tools for represent-
ing many problems in physics, engineering, economics, computational biology,
computerized medicine, to name but a few, which extend beyond optimization,
see [1] for the extensive review of the subject. Apart from the mathematical
problems connected with the characterization of solutions and development of
the appropriate algorithmic tools to nd them, modern problems oer signif-
icant implementation challenges due to their non-linearity and large scale. It
leaves just a few options for the algorithms development as it occurs in the oth-
ers related elds like convex feasibility (CF) problems [2] as well. One of these
options is to use xed point iteration methods with various attraction properties
toward the solutions, which have low memory requirements and simple and easily
parallelized iterations. These schemes are quite popular for convex optimization
and CF problems but they need certain modications to be applied to VI prob-
lems. The idea of modication can be related to some approaches put forward
for convex optimization and CF problems in [35] and which is becoming known
as superiorization technique (see also [6] for the general description).
From the point of view of this approach the optimization problem
min f (x), xX (1)

https://doi.org/10.1007/978-3-319-69404-7_15
Sharp Penalty Mappings 211
or VI problem to nd x X such that
F (x )(x x ) 0, xX (2)
are conceptually divided into the feasibility problem x X and the second-
stage optimization or VI problems next. Then these tasks can be considered to a
certain extent separately which makes it possible to use their specics to apply
the most suitable algorithms for feasibility and optimization/VI parts.
The problem is to combine these algorithms in a way which provides the
solution of the original problems (1) or (2). As it turns out these two tasks can
be merged together under rather reasonable conditions which basically require
a feasibility algorithm to be resilient with respect to diminishing perturbations
and the second-stage algorithm to be something like globally convergent over
the feasible set or its small expansions.
Needless to say that this general idea meets many technical diculties one of
them is to balance in intelligent way the feasibility and optimization/VI steps.
If optimization steps are essentially smaller than feasibility steps then it is
possible to prove general convergence results [3,4] under rather mild conditions.
However it looks like that this requirements for optimization steps to be smaller
(in fact even vanishing compared to feasibility) slows down the overall optimiza-
tion in (1) considerably.
This can be seen in the text-book penalty function method for (1) which
consists in the solution of the auxiliary problem of the kind
min{ X (x) + f (x) } = X (x ) + f (x ) (3)

x
where X (x) = 0 for x X and X (x) > 0 otherwise. The term f (x) can
be considered as the perturbation of the feasibility problem minx (x) and for
classical smooth penalty functions the penalty parameter > 0 must tend to
zero to guarantee convergence of x to the solution of (1). Denitely it makes
the objective function f (x) less inuential in solution process of (1) and hinders
the optimization.
To overcome this problem the exact penalty functions X () can be used
which provide the exact solution of (1)
min{ X (x) + f (x) } = f (x ) (4)

x
for small enough > 0 under rather mild conditions. The price for the conceptual
simplication of the solution of (2) is the inevitable non-dierentiability of the
penalty function X (x) and the corresponding worsening of convergence rates
for instance for gradient-like methods (see [8,9] for comparison). Nevertheless
the idea has a certain appeal, keeping in mind successes of nondiereniable
optimization, and the similar approaches with necessary modications were used
for VI problems starting from [10] and followed by [1114] among others. In these
works the penalty functions were introduced and their gradient elds direct the
iterations to feasibility.
212 E. Nurminski
Here we suggest a more general denition of a sharp penalty mapping P :

E C(E), not necessarily potential, which is oriented toward a feasible set (for
details of notations see the Sect. 2). It also admits a certain problem-dependent
penalty constant > 0 such that the sum F + P of variational operator F
of (2) and P scaled by possesses a desirable properties to make the iteration
algorithm converge at least to a given neighborhood of solution of (2). In the
preliminary form this idea was suggested in [15] using a dierent denition of a
sharp penalty mapping which resulted in rather weak convergence result. Here
we show that it is possible to reach stronger convergence result with all limit
points of the iteration method being an -solutions of VI problem (2).
2 Notations and Preliminaries
Let E denotes a nite-dimensional space with the inner product xy for x, y

E, and the standard Euclidean norm x = xx. The one-dimensional E is
denoted as R and R = R {}. The unit ball in E is denoted as B = {x :
x 1}. The space of bounded closed convex subsets of E is denoted as C(E).
The distance function (x, X) between point x and set X E is dened as
(x, X) = inf yX x y. The norm of a set X is dened as X = supxX x.
For any X E its interior is denoted as int(X), the closure of X is denoted
as cl(X) and the boundary of X is denoted as X = X \ int(X).
The sum of two subsets A and B of E is denoted as A + B and understood
as A + B = {a + b, a A, b B}. If A is a singleton {a} we write just a + B.
Any open subset of E containing zero vector is called a neighborhood of zero
in E. We use the standard denition of upper semi-continuity and monotonicity
of set-valued mappings:
Definition 1. A set-valued mapping F : E C(E) is called upper semi-
continuous if at any point x for any neighborhood of zero U there exists a neigh-
borhood of zero V such that F (x) F (x) + U for all x x + V .
Definition 2. A set-valued mapping F : E C(E) is called a monotone if

(fx fy )(x y) 0 for any x, y E and fx F (x), fy F (y).
We use standard notations of convex analysis: if h : E R is a convex

function, then dom(h) = {x : h(x) < }, epi h = {(, x) : h(x), x
dom(h)} R E, the sub-dierential of h is dened as follows:
Definition 3. For a convex function h : E R a sub-dierential of h at
point x dom(h) is the set h(x) of vectors g such that h(x) h(x) g(x x)
for any x dom(h).
This denes a convex-valued upper semi-continuous maximal monotone set-
valued mapping h : int(dom(h)) C(E). At the boundaries of dom(h) the
sub-dierential of h may or may not exist. For dierentiable h(x) the classical
gradient of h is denoted as h (x).
We dene the convex envelope of X E as follows.
Definition 4. An inclusion-minimal set Y C(E) such that X Y is called a

convex envelope of X and denoted as co(X).
Our main interest consists in nding a solution x of a following nite-
dimensional VI problem with a single-valued operator F (x):
Find x X C(E) such that F (x )(x x ) 0 for all x X. (5)
This problem has its roots in convex optimization and for F (x) = f (x) VI (5)
is the geometrical formalization of the optimality conditions for (1).
If F is monotone, then the pseudo-variational inequality (PVI) problem
Find x X such that F (x)(x x ) 0 for all x X. (6)
has a solution x which is a solution of (5) as well. However it is not necessary

for F to be monotone to have a solution of (6) which coincides with a solution
of (5) as Fig. 1 demonstrates.
Oriented but non-monotone operator F (x)

F (x) = x + 0.3x2 sin(25x)
F (x ) = 0
X = [1, 1]
Fig. 1. Non-monotone operator F (x) oriented toward x = 0.
For simplicity we assume that both problems (5) and (6) has unique and
hence coinciding solutions.
To have more freedom to develop iteration methods for the problem (6) we
introduce the notions of oriented and strongly oriented mappings according to
the following denitions.
214 E. Nurminski
Definition 5. A set-valued mapping G : E C(E) is called oriented toward x

at point x if
gx (x x) 0 (7)
for any gx G(x).
Definition 6. A set-valued mapping G : E C(E) is called strongly oriented
toward x on a set X if for any > 0 there is > 0 such that
gx (x x) (8)
for any gx G(x) and all x X \ {x + B}.

If G is oriented (strongly oriented) toward x at all points x X then we will
call it oriented (strongly oriented) toward x on X.
Of course if x = x , a solution of PVI problem (6), then G is oriented toward

x on X by denition and the other way around.
The notion of oriented mappings is somewhat related to attractive mappings
introduced in [2], which can be dened for our purposes as follows.
Definition 7. A mapping F : E E is called attractive with respect to x at
point x if
F (x) x x x (9)
It is easy to show that if F is an attractive mapping, then G(x) = F (x) x is
an oriented mapping, however G(x) = 10x is the oriented mapping toward {0}
on [1, 1] but neither G(x) nor G(x) + x are attractive.
Despite the fact that the problem (5) depends upon the behavior of F on X
only, we need to make an additional assumption about global properties of F
to avoid certain problems with possible divergence of iteration method due to
run-away eect. Such assumption is the long-range orientation of F which is
frequently used to ensure the desirable global behavior of iteration methods.
Definition 8. A mapping F : E E is called long-range oriented toward a set
X if there exists F 0 such that for any x X
F (x)(x x) > 0 for all x such that x F . (10)
We will call F the radius of long-range orientation of F toward X.
3 Sharp Penalty Mappings

In this section we present the key construction which makes possible to reduce
an approximate solution of VI problem into calculation of the limit points of
iterative process, governed by strongly oriented operators.
For this purpose we modify slightly the classical denition of a polar cone of
a set X.
Definition 9. The set KX (x) = {p : p(x y) 0 for all y X} we will call
the polar cone of X at a point x.
Of course KX (x) = {0} if x int X.

For our purposes we need also a stronger denition which denes a certain
sub-cone of KX (x) with stronger pointing toward X.
Definition 10. Let 0 and x
/ X + B. The set

KX (x) = {p : p(x y) 0 for all y X + B} (11)
will be called -strong polar cone of X at x.

As it is easy to see that the alternative denition of KX (x) is KX (x) = {p :
p(x y) p for all y X.}
To dene a sharp penalty mapping for the whole space E we introduce a
composite mapping

{0} if x X

KX (x) = KX (x) if x cl{{X + B} \ X} (12)

KX (x) if x F B \ {X + B}

Notice that KX (x) is upper semi-continuous by construction.
Now we dene a sharp penalty mapping for X as

PX (x) = {p KX (x), p = 1}. (13)

Clear that PX (x) is not dened for x int{X} but we can dened it to be equal

to zero on int{X} and take a convex envelope of PX (x) and {0} at the boundary
of X to preserve upper semi-continuity.

For some positive dene F (x) = F (x)+PX (x). Of course by construction
F (x) is upper semi-continuous for x / X.
For the further development we establish the following result on construction
of an approximate globally oriented mapping related to the VI problem (5).
Lemma 1. Let X E is closed and bounded, F is monotone and long-range
oriented toward X with the radius of orientability F and strongly oriented
toward solution x of (5) on X with the constants > 0 for > 0, satisfy-

ing (8) and PX () is a sharp penalty (13). Then for any suciently small > 0
there exists > 0 and > 0 such that for all a penalized mapping

F (x) = F (x) + PX (x) satises the inequality
fx (x x ) (14)
for all x F B \ {x + B} and any fx F (x).
Proof. For monotone F we can equivalently consider a pseudo-variational

inequality (6) with the same solution x . Dene the following subsets of E:
X = X \ {x + B},
(1)
X = {{X + B} \ X} \ {x + B},

(2)
(15)
X = F B \ {{X + B} \ {x + B}}.
(3)
216 E. Nurminski
Correspondingly we consider 3 cases.

(1)
Case A. x X . In this case f (x) = F (x) and therefore
f (x)(x x ) = F (x)(x x ) > 0. (16)
(2)
Case B. x X . In this case f (x) = F (x) + pX (x) where pX (x)
KX (x), pX (x) = 1 and therefore
f (x)(x x ) = F (x)(x x ) + pX (x)(x x ) /2 > 0. (17)
as pX (x)(x x ) > 0 by construction.
(3)
Case C. x X . In this case f (x) = F (x) + pX (x) where pX (x)

KX (x), pX (x) = 1. By continuity of F the norm of F is bounded on F B by

some M and as PX () is -strong penalty mapping
f (x)(x x ) = F (x)(x x ) + pX (x)(x x )
(18)
M x x + 2F M + F M > 0
for F M/.
By combining all three bounds we obtain
f (x)(x x ) min{ /2, F M } = > 0 (19)
for = F M/ which completes the proof.
The elements of a polar cone for a given set X can be obtained by dierent
means. The most common are either by projection onto set X:
x X (x) KX (x) (20)
where X (x) X is the orthogonal projection of x on X, or by subdierential
calculus when X is described by a convex inequality X = {x : h(x) 0}. If
there is a point x such that h(x) < 0 (Slater condition) then h(y) < 0 for all
y int{X}. Therefore 0 < h(x) h(y) gh (x)(x y) for any y int{X}.
By continuity 0 < h(x) h(y) gh (x)(x y) for all y X which means that
gh KX (x).
One more way to obtain gh KX (x) relies on the ability to nd some
xc int{X} and use it to compute Minkowski function
X (x, xc ) = inf { : xc + (x xc )1 X} > 1 for x
/ X. (21)
0
Then by construction x = xc + (x xc )X (x, xc )1 X, i.e. h(x) = 0 and for

any gh h(x) the inequality gh x gh y holds for any y X.
By taking y = xc obtain gh x gh xc and therefore
gh x = gh xc + gh (x xc )X (x, xc )1 = X (x, xc )1 gh x + (1 X (x, xc )1 )gh xc
(22)
X (x, xc )1 gh x + (1 X (x, xc )1 )gh x.
Hence gh x gh x gh y for any y X, which means that gh KX (x).

As for -expansion of X it can be approximated from above (included into)
by the relaxed inequality X + B {x : h(x) L} where L is a Lipschitz
constant in an appropriate neighborhood of X.
4 Iteration Algorithm
After construction of the mapping F , oriented toward solution x of (6) at the
whole space E except -neighborhood of x we can use it in an iterative manner
like
xk+1 = xk k f k , f k F (xk ), k = 0, 1, . . . , (23)
where {k } is a certain prescribed sequence of step-size multipliers, to get the
sequence of {xk }, k = 0, 1, . . . which hopefully converges under some conditions
to to at least the set X = x + B of approximate solutions of (5).
For technical reasons, however, it would be convenient to guarantee from the
very beginning the boundedness of {xk }, k = 0, 1, . . .. Possibly the simplest way
to do so is to insert into the simple scheme (23) a safety device, which enforces
restart if a current iteration xk goes too far. This prevents the algorithm from
divergence due to the run away eect and it can be easily shown that it keeps
a sequence of iterations {xk } bounded.
Thus the nal form of the algorithm is shown as the gure Algorithm 1,

assuming that the set X, the operator F and the sharp penalty mapping PX
satisfy conditions of the Lemma 1. We prove convergence of the Algorithm K 1
under common assumptions on step sizes: k +0 when k and k=1 k
when K . This is not the most ecient way to control the algorithm,
but at the moment we are interested mostly in the very fact of convergence.
Data: The variational inequality operator F , sharp penalty mapping PX ,

positive constant , penalty constant which satisfy conditions of the
Lemma 1, long-range orientation radius F , a sequence of step-size
multipliers {0 < k , k = 0, 1, 2, . . . }. and an initial point x0 F B.
Result: The sequence of approximate solutions {xk } where every converging
sub-sequence has a limit point which belongs to a set X of -solution
of variational inequality (5).
Initialization;
Define penalized mapping
F (x) = F (x) + PX (x), (24)

and set the iteration counter k to 0;
while The limit is not reached do
Generate a next approximate solution xk+1 :
k
x k f k , f k F (xk ), if xk 2F
xk+1 = (25)
x0 otherwise.
Increment iteration counter k k + 1;

end
Complete: accept {xk }, k = 0, 1, . . . as an approximate solution of (5).a
a
The exact meaning of this will be clarified in the convergence Theorem 1.
Algorithm 1. The generic structure of a conceptual version of the iteration
algorithm with exact penalty.
218 E. Nurminski
Theorem 1. Let > 0, , F, PX satisfyKthe assumptions of the Lemma 1, >

, and k +0 when k and k=1 k when K . Then all
limit points of the sequence {xk } generated by the Algorithm 1 belong to the set
of -solutions X = x + B of the problem (5).
Proof. We show rst the boundedness of the sequence {xk }. To do so it is
sucient to demonstrate that the sequence {xk , k = 1, 2, . . . } crosses the
interval [F , 2F ] a nite number of times only (any way, from below or from
above). Show rst that the sequence {xk } leaves the set 32 F B a nite number
of times only. Dene ( a nite or not ) set T of indices T = {tk , k = 1, 2, . . . }
such that
3 3
x < F and x +1 F . (26)
2 2
If T then
x +1 2 = x k F (x )2 = x 2 2 f x + 2 f 2
(27)
x 2 2 f x + 2 C 2 x 2 2 + 2 C 2 ,
where C is an upper bound for F (x) with x 2F B and > 0 is a lower
bound for f x for x 2F B and f F (x). For large enough < C 2
and hence
x +1 2 x 2 < x 2 (28)
which contradicts the denition of the set T . Therefore T is a nite set and the
sequence {xk } leaves the set 32 F B a nite number of times only which proves
the boundedness of {xk }.
Dene now W (x) = xx 2 and notice that due to the boundedness of {xk }
and semi-continuity of F (x) and etc., W (xk+1 ) W (xk ) 0 when k . It
implies that the limit set
W = {w : the sub-sequence {xks } exists such that lim W (xks ) = w } (29)
s
is a certain interval [wl , wu ] R+ and the statement of the theorem means that
wu 2 .
To prove this we assume contrary, that is wu > 2 and hence there exists a
sub-sequence {xks , s = 1, 2, . . . } such that lims W (xks ) = w > 2 . Without
loss of generality we may assume that lims xks = x and of course x / X .
Therefore f (x x ) > 0 for any f F (x ) and by upper semi-continuity of F
there exists an > 0 such that F (x)(x x ) for all x x + 4B
and some
> 0. Again without loss of generally we may assume that < ( w )/4 so
(x + 4B) (x + B) = .
For for s large enough xks x + B and let us assume that for all t > ks
the sequence {xt , t > ks } xks + B x + 2B.
Then
W (xt+1 ) = xt t F (xt ) x 2 = W (xt ) 2t F (xt )(xt x ) + t2 F (xt )2

W (xt ) 2t F (xt )(xt x ) + t2 C 2 W (xt ) 2t + t2 C 2 < W (xt ) t ,
(30)
for all t > ks and s large enough that supt>ks t < /C 2 . Summing up last
inequalities from t = ks to t = T 1 obtain
T
1
W (xT ) W (xks ) t (31)
t=ks
when T which is of course impossible.

Hence for each ks there exists rs > ks such that xks xrs > > 0 Assume
that rs is in fact a minimal such index, i.e. xks xt for all t such that
ks < t < rs or xt xtk + B x + 2B for all such t. Without any loss of
generality we may assume that xrs x where by construction x x > 0
and therefore x = x .
As all conditions which led to (31) hold for T = rs then by letting T = rs we
obtain
r
s 1
W (xrs ) W (xks ) t . (32)

t=ks
On the other hand

rs
r
s 1 rs

< xks xrs xt+1 xt t F (xt ) K t (33)
t=ks t=ks t=ks
where K is the upper estimate of the norm of F (x) on 2F B.

rs 1
Therefore t=k s
t > /K > 0 and nally
W (xrs ) W (xks ) /K. (34)
Passing to the limit when s obtain W (x ) W (x ) /K < W (x ) Also
W (x ) > 2 as x x + 4B which does not intersect with x + B. To save on
notations denote W (x ) = w and W (x ) = w .
In other words, assuming that w > 2 we constructed another limit point
w of the sequence {W (xk )} such that 2 < w < w . It follows from this that

the sequence {W (xk )} innitely many times crosses any sub-interval [w , w ]
(w , w ) both in up and down directions and hence there exist sub-sequences
{ps , s = 1, 2, . . . } and {qs , s = 1, 2, . . . } such that ps < qs and
W (xps ) w , W (xqs ) w , W (xt ) (w , w ) for ps < t < qs (35)
Then
q
s 1
0 < W (xqs ) W (xps ) = (W (xt+1 ) W (xt )) (36)

t=ps
and hence for any s there is an index ts : ps < ts < qs such that
0 < W (xts +1 ) W (xts ). (37)

220 E. Nurminski
However as W (xts ) > w , xts

/ x + B and therefore
W (xts +1 ) W (xts ) = xts +1 x 2 xts x 2 =

xts x + ts f ts 2 xts x 2 = (38)
2ts f (x x ) + t2s f ts 2 = ts (2f ts (xts x ) + ts f ts 2 ),
ts ts
where f ts F (xts ). Notice that f ts (xts x ) < > 0 and f ts 2 C. Using

these estimates we obtain
W (xts +1 ) W (xts ) ts (2 ts C) ts < 0 (39)
for all s large enough. This contradicts (37) and therefore proves the
theorem.
5 Conclusions
In this paper we dene and use a sharp penalty mapping to construct the itera-
tion algorithm converging to an approximate solutions of monotone variational
inequalities. Sharp penalty mappings are analogues of gradient elds of exact
penalty functions but do not need to be potential mappings. Three examples of
sharp penalty mappings are given with one of them seems to be a new one. The
algorithm consists in recursive application of a penalized variational inequal-
ity operator, but scaled by step-size multipliers which satisfy classical diverging
series condition. As for practical value of these result it is generally believed
that the conditions for the step-size multipliers used in this theorem result in
rather slow convergence of the order O(k 1 ). However the convergence rate can
be improved by dierent means following the example of non-dierentiable opti-
mization. The promising direction is for instance the least-norm adaptive reg-
ulation, suggested probably rst by A. Fiacco and McCormick [16] as early as
1968 and studied in more details in [17] for convex optimization problems. With
some modication in can be easily used for VI problems as well. Experiments
show that under favorable conditions it produces step multipliers decreasing as
geometrical progression which gives a linear convergence for the algorithm. This
may explain the success of [7] where geometrical progression for step multipliers
was independently suggested and tested in practice.
Acknowledgements. This work is supported by the Ministry of Science and Educa-

tion of Russian Federation, project 1.7658.2017/6.7
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processes with attractants. Optim. Methods Softw. 25(1), 97108 (2010)
A Nonconvex Optimization Approach
to Quadratic Bilevel Problems
Andrei Orlov(B)
Matrosov Institute for System Dynamics and Control Theory of Siberian

Branch of RAS, Irkutsk, Russia
anor@icc.ru
http://nonconvex.isc.irk.ru
Abstract. This paper addresses one of the classes of bilevel optimiza-

tion problems in their optimistic statement. The reduction of the bilevel
problem to a series of nonconvex mathematical optimization problems,
together with the specialized Global Search Theory, is used for devel-
oping methods of local and global searches to find optimistic solutions.
Illustrative examples show that the approach proposed is prospective
and performs well.
Keywords: Bilevel optimization Quadratic bilevel problems Opti-

mistic solution KKT-approach Global Search Theory Computational
simulation
1 Introduction
As well-known, problems with hierarchical structure arise in investigations of com-
plex control systems [9] with the bilevel optimization being the most popular
modeling tool [7]. According to Pang [17], a distinguished expert in optimization,
development of methods for solving various problems with hierarchical structure
is one the three challenges faced by optimization theory and methods in the 21st
century. This paper investigates one of the classes of bilevel problems with a con-
vex quadratic upper level goal function and a quadratic lower level goal function if
the constraints are linear. The lower level goal function has a bilinear component.
The task is to nd an optimistic solution in the situation when the actions of the
lower level might coordinate with the interests of the upper level [7].
During more than three decades of intensive investigation of bilevel optimiza-
tion problems, many methods for nding the optimistic solutions were proposed
by dierent authors (see the surveys [6,8]). Nevertheless, as far as we can con-
clude on the basis of the available literature, a few results published so far deal
with numerical solutions of merely test bilevel high-dimension problems (for
example, up to 100 variables at each level for linear bilevel problems [19]). Most
frequently authors consider just illustrative examples with the dimension up to
10 (see [14,18]) and only the works [1,5,10,12] present some results on solving
nonlinear bilevel problems of dimension up to 30 at each level.
https://doi.org/10.1007/978-3-319-69404-7_16
A Nonconvex Optimization Approach to Quadratic Bilevel Problems 223
At the same time, in our group we have an experience of solving linear bilevel
problems with up to 500 variables at each level [11] and quadratic-linear bilevel
problems of dimension up to (150 150) [23]. Here we generalize our methods
for problems with quadratic goal functions at each level.
For this purpose, we use the most common approach to address a bilevel prob-
lem via its reduction to a single-level one by replacing the lower level problem
with optimality conditions (the so called KKT-approach) [7,23]. Then, using the
penalty method, the resulting problem with a nonconvex feasible set is reduced
to a series of problems with a nonconvex function under linear constraints [7,23].
The latter ones turn out to be the d.c. optimization problems (with goal func-
tions that can be represented as the dierence of two convex functions) [20,21],
which can be addressed by means of the Global Search Theory developed by
Strekalovsky [20,21]. In contrast to the generally accepted global optimization
methods such as branch-and-bound based techniques, approximation methods
and the like, this theory employs reduction of the nonconvex problem to a family
of simpler problems with a possibility of application of classic convex optimiza-
tion methods.
In accordance with the Global Search Theory, this paper aims at constructing
specialized local and global search methods for nding optimistic solutions to the
problems under study. Illustrative examples taken from the literature were used
to demonstrate that the approach proposed for numerical solution of quadratic
bilevel problems performs rather well.
2 Statement of the Problem and Its Reduction
Consider the following quadratic-quadratic problem of bilevel optimization in its

optimistic statement. In this case, according to the theory [7], at the upper level
we perform the minimization with respect to the variables of both levels which
are in cooperation:
1 1
F (x, y) := x, Cx + c, x + y, Dy + d, y min,

2 2 x,y

x X := {x IRm |Ax b},
1 (QBP)
y Y (x) := Arg min{ y, D1 y + d1 , y + x, Qy| y Y (x)},

y 2

Y (x) ={y IRn |A1 x + B1 y b1 },
where A IRpm , A1 IRqm , B1 IRqn , C IRmm , D, D1 IRnn ,

Q IRmn , c IRm , d, d1 IRn , b IRp , b1 IRq . Additionally, C = C T 0,
1
D = DT 0, D1 = D1T 0. The quadratic terms of the form x, C1 x+c1 , x,
2
where C1 = C1T 0, are not included into the lower level goal function, because
for a xed upper level variable x they are constant and do not aect the structure
of the set Y (x).
224 A. Orlov
Note that despite the bilinear component in the goal function, for a xed
x X the lower level problem
1
y, D1 y + xQ + d1 , y min, A1 x + B1 y b1 (FP(x))
2 y
is convex with automatically fullled Abadie regularity conditions [2] due to

ane constraints. Therefore, the Karush-Kuhn-Tacker conditions [2] are nec-
essary and sucient for Problem (FP(x)). According to these conditions, the
point y is a solution to (FP(x)) (y Sol(FP(x))) if and only if there exists a
vector v IRq such that

D1 y + d1 + xQ + vB1 = 0, v 0, A1 x + B1 y b1 ,
(KKT )
v, A1 x + B1 y b1 = 0.
By replacing the lower level problem in (QBP) with its optimality conditions
(KKT ) we obtain the following mathematical optimization problem:

F (x, y) min , Ax b,

x,y,v
D1 y + d1 + xQ + vB1 = 0, v 0, A1 x + B1 y b1 , (DCC)
v, A1 x + B1 y b1 = 0.
It is clear that (DCC) is a problem with a nonconvex feasible set because the
nonconvexity is generated by the last equality constraint (complementary con-
straint). The following theorem on the equivalence between (QBP) and (DCC)
is valid.
Theorem 1 [7]. For the pair (x , y ) to be a global solution to (QBP) it is
necessary and sucient that there exists a vector v IRq such that the triple
(x , y , v ) is a global solution to (DCC).
This theorem reduces the search for an optimistic solution to the bilevel
problem (QBP) to solving Problem (DCC). Note that direct handling of the
nonconvex set in this problem is quite challenging, which is why we use the
penalty method to boil it down to a series of nonconvex problems with a convex
feasible set.
It is easy to see that the complementary constraint can be written in the
equivalent form v, b1 A1 x B1 y = 0. Then (x, y, v) D, where
D := {(x, y, v) | Ax b, D1 y + d1 + xQ + vB1 = 0, v 0, A1 x + B1 y b1 }
is the convex set in IRn+m+q , the following inequality holds v, b1

A1 x B1 y 0. Now the penalized problem can be written as

1 1
(x, y, v) := x, Cx + c, x + y, Dy + d, y+
2 2 (DC())
+ v, b1 A1 x B1 y min , (x, y, v) D,
x,y,v
where > 0 is a penalty parameter. For a xed this problem belongs to the
class of d.c. minimization problems [20,21] with a convex feasible set. In what
follows we show that the goal function of (DC()) can be represented as a dif-
ference of two convex functions. Let (x(), y(), v()) be a solution to (DC())
for some . Denote r[] := v(), b1 A1 x() B1 y() and formulate the fol-
lowing result on the connection between the solutions to (DCC) and (DC()).
Proposition 1 [2,7]
(i) Let for some > 0 the equality r[] = 0 holds for a solution (x(), y(),
v()) to Problem (DC()). Then, the triple (x(), y(), v()) is a solution
to Problem (DCC).
(ii) For all values of the parameters > the function r[] vanishes, so that
the triple (x(), y(), v()) is a solution to Problem (DCC).
Therefore, the established connection between Problems (DC()) and (DCC)

enables us to search for a solution to Problem (DC()) instead of a solution to
Problem (DCC), when the parameter grows. It can be shown that there exists a
nite value of with r[] = 0 [3]. For a xed , we will address Problem (DC())
by means of the Global Search Theory developed by Strekalovsky [20,21]. Within
that theory, it is rst required to construct a local search procedure that takes
into consideration special features of the problem under study.
3 The Local Search

It is noteworthy that nonconvexity in Problem (DC()) is generated by only a
bilinear component v, b1 A1 x B1 y. On the basis of this fact, we suggest
to perform the local search in Problem (DC()) using the idea of the succes-
sive solution by dierent groups of variables. Earlier this idea was successfully
applied in solving bimatrix games [16,22], problems of bilinear programming
[15,22], and quadratic-linear bilevel problems [23]. Even though, contrarily to
the problems mentioned above, the feasible set D is not split up into dierent
groups of variables, Problem (DC()) turns into a convex quadratic optimiza-
tion problem for a xed v; meanwhile we obtain the linear programming (LP)
problem with respect to the variable v for a xed pair (x, y).
Therefore, a specialized local search method appears. Let there be given a
starting point (x0 , y0 , v0 ).
V -procedure
s
Step 0. Set s := 1, v := v0 .
s
Step 1. Using a suitable quadratic programming method, nd the -
2
solution (xs+1 , y s+1 ) to the problem

1 1
x, Cx + c, x + y, Dy + d, y v s A1 , x v s B1 , y min,
2 2 x,y (QP(v s ))
Ax b, A1 x + B1 y b1 , D1 y + d1 + xQ + v s B1 = 0.
226 A. Orlov
s
Step 2. Find the -solution v s+1 to the following LP-problem:
2

b1 A1 xs+1 B1 y s+1 , v min,
v (LP(xs+1 , y s+1 ))
D1 y s+1 + d1 + xs+1 Q + vB1 = 0, v 0.
Step 3. Set s := s + 1 and move to Step 1.

Note that the name of the V-procedure was not chosen randomly. To
launch the algorithm, we do not need all the components of the starting point
(x0 , y0 , v0 ), knowing only v0 is sucient. Herein, it is necessary to choose the
components (x0 , y0 , v0 ) so that the auxiliary problems of linear and quadratic
programming were solvable at steps of the algorithm. The solvability can be guar-
anteed, for example, by an appropriate choice of the feasible point (x0 , y0 , v0 )
D. It also should be noted that here, contrarily to the problems with disjoint
constrains, local subproblems are each time solved on new sets that depend on
the components of the current point. Nonetheless, it happened to be possible to
prove the following convergence theorem for the V -procedure.
Theorem 2

(i) Let a sequence {s } be such that s > 0, s = 1, 2, ..., and s=1 s <
+. Then the number sequence {s := (xs , y s , v s )} generated by the V -
procedure converges.
(ii) If (xs , y s , v s ) (x, y, v), then the accumulation point (x, y, v) satises the
inequalities
(x, y, v) (x, y, v) (x, y) D(v), (1)
(x, y, v) (x, y, v) v D(x, y), (2)
where D(v) := {(x, y)| (x, y, v) D}, D(x, y) := {v| (x, y, v) D}.
Definition 1. The triple (x, y, v) satisfying (1) and (2) is said to be the critical
point in Problem (DC()). If the inequalities (1) and (2) are satised with a
certain accuracy then we refer to this point as approximately critical.
It can be shown (see, e.g., [23]), that if we use, for example, the following
inequality as a stopping criterion for the V -procedure
s s+1 , (3)
where is a given accuracy, then after the nite number of iterations of the local
search method we arrive at the approximately critical point. Recall that such
a denition of critical points is quite advantageous when we perform a global
search in problems with a bilinear structure [22,23]. The next section describes
basic elements of the global search for Problem (DC()).
4 Global Optimality Conditions and the Global Search

Procedure
As well-known, the local search does not provide, in general, a global solution in
nonconvex problems of even moderate dimension [2022]. Therefore, in what fol-
lows we discuss the procedure of escaping critical points obtained during the local
search. The procedure is based on the Global Optimality Conditions (GOCs)
developed by Strekalovsky for the d.c. minimization problems [20,21]. To build
the global search procedure, rst of all we need an explicit d.c. representation of
the goal function of (DC()). We will employ the following representation based
on the known property of scalar product:
(x, y, v) = g(x, y, v) h(x, y, v), (4)

1 1 2
where g(x, y, v) = x, Cx+c, x+ y, Dy+d, y+b1 , v+ A1 x v +
2 2 4
2 2 2
B1 y v , h(x, y, v) = A1 x + v + B1 y + v . Note that the so-called
4 4 4
basic nonconvexity in Problem (DC()) is provided by the function h (for more
detail, refer to [20]).
The necessary Global Optimality Conditions that constitute the basis of the
global search procedure have the following form in terms of Problem (DC()).
Theorem 3 [20,21,23]. If the feasible point (x , y , v ) is a (global) solution to

Problem (DC()), then (z, u, w, ) Rm+n+q+1 :
h(z, u, w) = , := (x , y , v ), (5)
g(x, y, v) sup (g, D), (6)

x,y,v
g(x, y, v) h(z, u, w), (x, y, v) (z, u, w) (x, y, v) D. (7)
The conditions (5)(7) possess the so-called algorithmic (constructive) prop-

erty: if the GOCs are violated, we can construct a feasible point that will be
better than the point in question [20,21]. Indeed, if for some (z, u, w, ) from
(5) on some level := k := (xk , y k , v k ) for the feasible point (x, y, v) D the
inequality (7) is violated:
g(x, y, v) < + h(z, u, w), (x, y, v) (z, u, w) ,
then it follows from the convexity of h() that
(x, y, v) = g(x, y, v) h(x, y, v) < h(z, u, w) + h(z, u, w) = (xk , y k , v k ),
or, (x, y, v) < (xk , y k , v k ). Therefore, the point (x, y, v) D happens to be

better with respect to the goal function value than the point (xk , y k , v k ). Con-
sequently, by varying parameters (z, u, w, ) in (5) for a xed = k and nding
228 A. Orlov
approximate solutions (x(z, u, w, ), y(z, u, w, ), v(z, u, w, )) of the linearized

problems (see (7))
g(x, y, v) h(z, u, w), (x, y, v) min (x, y, v) D, (PL(z, u, w))
x,y,v
we obtain a family of starting points to launch the local search procedure.

Additionally, we do not need to sort through all (z, u, w, ) at each level ,
because it is sucient to prove that the inequality (7) is violated at the single
4-tuple (z, u, w, ). After that we move to the new level (xk+1 , y k+1 , v k+1 ) :=
(x, y, v), k+1 := (xk+1 , y k+1 , v k+1 ) and vary parameters again.
Hence, according to the Global Search Theory developing by Strekalovsky
[20,21], Problem (DC()) can be split into several simpler problems (linearized
problems and problems with respect to other parameters from the global opti-
mality conditions), which can be represented in the form of the following Global
Search Strategy.
Suppose we know some approximately critical point (xk , y k , z k ) in Problem
(DC()) with the goal function value k := (xk , y k , v k ). The point has been
constructed with the help of a specialized local search method (for example, we
could use the V -procedure). Then we perform the following chain of operations.
(1) Choose a number [ , + ], where := inf(g, D), + := sup(g, D). We
can take, for example, g(xk , y k , z k ) as a starting value of the parameter
[20,21].
(2) Furtherfore, construct some nite approximation
Ak = {(z i , ui , wi ) | h(z i , ui , wi ) = k , i = 1, ..., Nk }
for the level surface U (k ) = {(x, y, v) | h(x, y, v) = k } of the convex
function h().
(3) For all approximation points Ak verify the inequality
g(z i , ui , wi ) , i = 1, 2, ..., Nk , (8)
that follows from the global optimality conditions for Problem (DC()) (see
(5)). If the inequality (8) is satised, then the approximation point will be
used in process. Otherwise, the point (z i , ui , wi ) is useless, because it is not
able to improve the current point [2023].
(4) For each point (z i , ui , wi ), i {1, 2, ..., Nk } chosen at Stage (3) nd approx-
imate solutions (z i , ui , wi ) of the linearized (with respect to the basic non-
convexity) Problems (PL(z i , ui , wi )).
(5) Using the points (z i , ui , wi ), perform the local search that delivers approxi-
mately critical points (xi , y i , v i ), i {1, ..., N } in Problem (DC()).
(6) For the chosen i {1, . . . , Nk }, solve the level problem:

x h(z, u, w), xi z + y h(z, u, w), y i u
+ v h(z, u, w), v i w max , h(z, u, w) = k . (Ui )
(z,u,w)
Note that denition for h() (see (4)) makes it possible to solve Problem (Ui )
analytically. Let (z0i , ui0 , w0i ) be the approximate solution to this problem.
(7) If for some j {1, . . . , Nk } the following inequality holds
g(xj , y j , v j ) < x h(z0j , uj0 , w0j ), xj z0j

+ y h(z0j , uj0 , w0j ), y j uj0 + v h(z0j , uj0 , w0j ), v j w0j ,
then, due to convexity of h(), we obtain
h(z0j , uj0 , w0j ) = k = (xk , y k , v k ) > (xj , y j , v j ).
Thus, we constructed the point (xj , y j , v j ) D, which is better than (xk , y k , v k ).

If we failed to improve the value of k using all approximation points Ak , then
we have to continue the one-dimensional search with respect to on the segment
[ , + ].
Observe that the key moment of the global search strategy described above
is that on Stage (2), when we construct an approximation for the level surface of
the convex function h() that denes the basic nonconvexity in Problem (DC()).
Very important to note that the approximation should be representative enough
to nd out whether the current critical point (xk , y k , z k ) is a global solution
or not [20,21]. To construct this approximation, we can employ, for example,
special direction sets that include, in particular, the Euclidean basis vectors and
the components of the current critical point (for more detail, see [2023]).
5 Computational Simulation
To illustrate the operability of the newly constructed local and global search
algorithms for nding optimistic solutions to quadratic bilevel problems, a few
examples of moderate dimension were taken from the available literature. Note
that here they have the form (QBP), with constant components excluded from
the goal functions on the upper and lower levels.
Example 1 ([18])

F (x, y) = x2 10x + 4y 2 + 4y min,
x,y

x X = {x IR1 | x 0},
y Y (x) = Arg min{y 2 2y | y Y (x)},

y

Y (x) = {y IR | 3x + y 3, x 0.5y 4, x + y 7, y 0}.
1
It is well-known that the global optimistic solution to this problem with the
goal function optimal value F = 9 is achieved at the point (x , y ) = (1, 0).
Example 2 ([1])

F (x, y) = x2 10x + 4y 2 + 4y min,
x,y

x X = {x IR1 | x 0},
y Y (x) = Arg min{y 2 2y 1.5xy | y Y (x)},

y

Y (x) = {y IR | 3x + y 3, x 0.5y 4, x + y 7, y 0}.
1
230 A. Orlov
Interestingly, this problem diers from the previous one only by having a
bilinear component in the lower level goal function. Obviously, it does not result
in noncovexity of the problem at the lower level, because for a xed x the bilin-
ear component becomes linear. Moreover, the global optimistic solution to this
problem with the goal function optimal value F = 9 is achieved at the same
point (x , y ) = (1, 0) as above.
Example 3 ([4])

1 1
F (x, y) = x2 x + y 2 min,

2 2 x,y

x X = {x IR1 | x 0},
1
y Y (x) = Arg min{ y 2 xy | y Y (x)},

y 2

Y (x) = {y IR1 | x y 1, x + y , x y 1.
This problem has a form of the kernel problems used for generating quadratic
bilevel problems by means of the method constructed in [4]. Depending on the
value of the parameter , the kernel problems are divided into following classes:
Class 1 ( = 1): the global optimistic solution F = 0.5 is achieved at the

point (x , y ) = (1, 0).
Class 2 ( = 1.5): the global solution F = 0.4375, (x , y ) = (1.25, 0.25).
Class 3 ( = 2): F = 0.25, (x , y ) = (0.5, 0.5) or (x , y ) = (1.5, 0.5).
Class 4 ( = 3): F = 0.25, (x , y ) = (0.5, 0.5).
Even though the problems discussed above dier from each other only by
a single parameter, they all have dierent properties and a dierent number of
local and global solutions (for more detail, refer to [4]).
Example 4 ([14])

F (x, y) = 7x1 + 4x2 + y12 + y32 y1 y3 4y2 min,

x,y

x X = {x IR2 | x1 + x2 1, x1,2 0, },
1 1
y Y (x) = Arg min{y1 3x1 y1 + y2 + x2 y2 + y12 + y22 + y32 + y1 y2 |

y 2 2

y Y (x)}, Y (x) = {y IR3 |x1 2x2 + 2y1 + y2 y3 + 2 0, y1,2,3 0}.
Apparently, this problem is more complex than the previous ones, because
the dimension of the upper level variable is 2, whereas the dimension of the
lower level variable is 3. The global optimistic solution approximately equals
F = 0.6426 and is achieved at the point (x , y ) = (0.609, 0.391, 0, 0, 1.828).
First, for each of the problems we write down its single-level equivalent. Note
that the dimension of the single-level problems is bigger than that one of the
original problems by exactly the number of constraints in the lower level of
the bilevel problem. Further, we penalize the complementary constraint in each
problem and afterwards apply the local and global search methods described
above.
The software that implements the methods developed was coded in MATLAB
7.11.0.584 R2010b [13]. The auxiliary problems of linear and convex quadratic
programming were solved by the standard MATLAB subroutines linprog and
quadprog with the default settings [13]. To run the software, we used the
computer with Intel Core i5-2400 processor (3.1 GHz) and 4 Gb RAM.
To construct feasible starting points for the local search method, we used the
projection of the chosen infeasible point (x0 , y 0 , v 0 ) = (0, 0, 0) onto the feasible
set D by solving the following problem:

1
(x, y, v) (x0 , y 0 , v 0 )2 min ,
2 x,y,v (PR(x0 , y 0 , v 0 ))
(x, y, v) D.
The solution to Problem (PR(x0 , y 0 , v 0 )) was taken as a starting point

(x0 , y0 , v0 ). The inequality (3) with = 105 was the stopping criterion for
the local search method.
The value of the penalty parameter was set as = 10. Besides, similar
to solving quadratic-linear bilevel problem [23], we need two extra parameters
to launch the global search algorithm. The rst parameter will be required at
Stage (3) of the global search, when, using the inequality (8), we verify whether
the point is suitable for future use. During numerical implementation we need
the parameter , which can be varied to change the accuracy with which the
inequality is satised (to reduce the impact of the round-o errors, see also
[15,16,22]). Thus, the global search algorithm will not utilize the points that fail
to satisfy the inequality
g(z i , ui , wi ) + , (9)
If we set, for example, = 0.0 then the algorithm works fast but is not always
ecient. When we increase , the algorithms quality improves. However, the
run time increases too, because the condition (9) gets relaxed and the number
of level surface approximation points to be investigated, grows.
Various values of the second parameter M are responsible for splitting the
segment [ , + ] into a corresponding number of subsegments to perform the
passive one-dimensional search with respect to . The segment lower bound
was set to equal 0, meanwhile the upper bound was estimated by + =
(m + n + q). When we increase the value of M , the algorithms accuracy grows,
of course, but it happens at the expense of the proportional increase of the run
time. At the initial stage the values of these parameters were chosen as = 0.0,
M = 2.
To approximate the level surface at Stage (2) of the global search procedure,
we used the following three sets which represent direction set variations from [23]:
Dir1 = {((x, y) el , v ej ), l = 1, ..., m + n, j = 1, ..., q};
Dir2 = {(el , ej ), l = 1, ..., m + n, j = 1, ..., q};

Dir3 = {(el , ej ), l = 1, ..., m + n, j = 1, ..., q}.
232 A. Orlov
Here el IRm+n , ej IRq are the Euclidean basis vectors of the corresponding
dimension, (x, y, v) is a current critical point. These sets proved to be most
ecient when solving test problems. Computational results are given in Table 1
with the following denotations:
N o is a number of example;
Dir is the most ecient direction set which delivered its global optimistic
solution for the given problem;
Loc stands for the number of start-ups of the local search procedure required
to nd the approximate global solution to the problem;
LP is the number of the LP problems solved during the operation of the
program;
QP stands for the number of auxiliary convex quadratic problems solved;
GIt is the number of iterations of the global search method (the number of
improved critical points obtained during the operation of the program);
(x , y ) is an optimal solution to the bilevel problem;
v is an optimal value of the auxiliary variables;
F = is the optimal value of the goal functions of Problems (QBP) and
(DC());
T is the operating time of the program (in seconds).
Table 1. Testing of the global search method
N o Dir Loc LP QP GIt (x , y ) v F = T

1 Dir3 10 11 21 2 (1; 0) (2; 0; 0; 0) 9.0 0.21
2 Dir3 10 11 21 2 (1; 0) (3.5; 0; 0; 0) 9.0 0.22
3.1 Dir1 11 13 24 3 (1; 0) (0; 1; 0) 0.5 0.23
3.2 Dir2 16 21 37 3 (1.25; 0.25) (0; 1; 0) 0.4375 0.29
3.3 Dir1 8 11 19 1 (0.5; 0.5) (0; 0; 0) 0.25 0.22
3.4 Dir1 8 10 18 1 (0.5; 0.5) (0; 0; 0) 0.25 0.21
4 Dir3 60 120 180 6 (0.602; 0.398; 0; 0; 1.807) (1.81; 2.81; 3.2; 0) 0.6396 0.94
First of all note that the values of the parameters , , and M specied above
happened to be insucient to solve Problem 4. To nd the global optimistic
solution in it, we needed the values = 15, = 0.1, M = 3. On the other hand,
we managed to somewhat improve the approximate solution against the results
from [14]. This involved 6 global search iterations and about 1 s of operating
time.
Analysis of the rest of results in the table shows that in all one-dimensional
problems the known global optimistic solutions at each level were found in less
than 0.3 s, which required between 1 and 3 iterations of the global search method
(for GIt = 1, the solution was obtained already at the local search stage).
As expected, Problem 4 happened to be more complex than the others, pri-
marily due to its dimension. This is attested by the values in columns Loc, LP ,
QP , and GIt, which can be considered as some complexity measure for the
problem under study.
Therefore, computational experiments demonstrated that the global search
theory performs well when applied to bilevel problems of quadratic optimization,
whereby varying of the parameters in the algorithm opens up great prospects
for solving simple as well as more complex problems.
6 Conclusion
This paper proposes an innovative approach to solving quadratic bilevel problems
based on their reduction to parametric problems of d.c. minimization with a
subsequent application of the Global Search Theory [20,21]. The specialized local
and global search methods have been constructed to nd optimistic solutions to
bilevel problems. The methods have proved to be ecient in solving test problems
of moderate dimension.
Further research suggests extension of the range and dimension of problems.
For this purpose, it is planned to implement a special method of generating test
problems of bilevel optimization from [4].
The results of numerical testing as well as our previous computational expe-
rience [11,15,16,22,23] allow us to expect that the approach proposed will prove
eective in solving quadratic bilevel problems of high dimension (probably, up to
100 100) with a supplementary possibility of exploiting modern software pack-
ages for solving auxiliary LP and convex quadratic problems (IBM CPLEX,
FICO Xpress etc.).
Acknowledgments. This work has been supported by the Russian Science Founda-
tion (Project no. 15-11-20015).
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An Experimental Study of Adaptive Capping
in irace
Leslie Perez Caceres1(B) , Manuel Lopez-Ibanez2 , Holger Hoos3 ,

and Thomas Stutzle1
1
IRIDIA, Universite Libre de Bruxelles, Brussels, Belgium
{leslie.perez.caceres,stuetzle}@ulb.ac.be
2
Alliance Manchester Business School, University of Manchester, Manchester, UK
manuel.lopez-ibanez@manchester.ac.uk
3
Computer Science Department, University of British Columbia, Vancouver, Canada
hoos@cs.ubc.cs
Abstract. The irace package is a widely used for automatic algorithm

conguration and implements various iterated racing procedures. The
original irace was designed for the optimisation of the solution quality
reached within a given running time, a situation frequently arising when
conguring algorithms such as stochastic local search procedures. How-
ever, when applied to conguration scenarios that involve minimising the
running time of a given target algorithm, irace falls short of reaching the
performance of other general-purpose conguration approaches, since it
tends to spend too much time evaluating poor congurations. In this
article, we improve the ecacy of irace in running time minimisation by
integrating an adaptive capping mechanism into irace, inspired by the one
used by ParamILS. We demonstrate that the resulting iracecap reaches
performance levels competitive with those of state-of-the-art algorithm
congurators that have been designed to perform well on running time
minimisation scenarios. We also investigate the behaviour of iracecap in
detail and contrast dierent ways of integrating adaptive capping.
1 Introduction
Algorithm conguration is the task of nding parameter settings (a congura-
tion) of a target algorithm that achieve high performance for a given class of
problem instances [6,8]. The appropriate choice of parameter settings is often
crucial for obtaining good performance, particularly when dealing with compu-
tationally challenging (e.g., N P-hard) problems. This choice usually depends
on the set or distribution of problem instances to be solved as well as on the
execution environment. Therefore, using appropriately chosen parameter values
is not only essential for reaching peak performance, but also for conducting fair
performance comparisons between dierent algorithms for the same problem.
Traditionally, algorithm conguration has been performed manually, relying
on experience and intuition about the behaviour of a given algorithm. However,
typical manual conguration processes are time-consuming and tedious; further-
more, they often leave the performance potential of a given target algorithm
https://doi.org/10.1007/978-3-319-69404-7_17
236 L.P. Caceres et al.
unrealised. In light of this, several automated algorithm conguration approaches

have been developed and are now used increasingly widely.Prominent examples
of general-purpose algorithm conguration procedures include ParamILS [13],
SMAC [12], GGA++ [1] and irace [4,17,18]. The key idea behind these and
other conguration procedures is to view algorithm conguration as a stochas-
tic optimisation problem that can be solved by eectively searching the space
of congurations of a given target algorithm A. The performance metrics most
commonly optimised in this context are the solution quality reached by A within
a certain time budget and the running time of A for nding a solution (of a cer-
tain quality) to a given problem instance.
The irace software is an automatic congurator based on the iterated F-race
procedure [4,7] and recent improvements [17,18]. It was initially developed for
conguring metaheuristic algorithms that optimise solution quality. In contrast,
minimisation of the running time of a given target algorithm was a major focus in
the development of ParamILS and SMAC, and both of them include an adaptive
capping [13] mechanism that is specically designed to improve eciency when
dealing with this performance objective. The key idea behind adaptive capping is
to reduce the time wasted in the evaluation of poorly performing congurations
by bounding the maximum running time permitted for each such evaluation. This
bound is calculated based on the best-performing conguration found so far. The
use of adaptive capping allows the congurator to prune poorly performing target
algorithm congurations early and to quickly focus the conguration budget on
promising areas of the space of congurations being searched.
In this work, we improve the ecacy of irace on algorithm conguration sce-
narios involving running time minimisation. We adapt the ideas of the adaptive
capping mechanism into the underlying iterated racing procedure and dene
an additional dominance criterion based on the performance of the elite con-
gurations obtained by irace. As a rst step, we show that by extending irace
with adaptive capping, resulting in our new iracecap method, we can signicantly
increase its performance on well-known and dicult conguration scenarios. An
additional analysis of various parameters of iracecap gives further insights into
the importance of the statistical testing procedures and other aspects of irace.
A nal comparison with other state-of-the-art conguration procedures for run-
ning time minimisation, namely ParamILS and SMAC, shows that iracecap
reaches highly competitive performance and, thus, broadens the range of cong-
uration scenarios for which irace can be considered a possible method of choice.
The remainder of this article is structured as follows. First, we describe irace
and the adaptive capping mechanism used by ParamILS (Sects. 2 and 3). Next,
in Sect. 4, we describe how we integrated adaptive capping into irace, and we
experimentally analyse the resulting iracecap in Sect. 5. In Sect. 6, we compare
iracecap to state-of-the-art congurators for minimising running time, and we
conclude in Sect. 7.
An Experimental Study of Adaptive Capping in irace 237
2 Elitist Iterated Racing in irace

irace is an iterated racing procedure [7] for automatic algorithm conguration.
It explores the parameter space of a target algorithm by iteratively sampling
parameter congurations and applying a racing procedure to select the best-
performing congurations. The racing procedure considers a sequence of problem
instances on which the candidate congurations are evaluated. At each stage of
the race, all candidate congurations are run on a specic problem instance; at
the end of the stage, congurations that perform statistically worse than others
are eliminated from the race, while all others proceed to the next stage. Once a
race is terminated, the best congurations, called elite, are used to update the
sampling model from which new congurations are generated. The elite congu-
rations are carried over to the next iteration to continue their evaluation within a
new race together with the newly generated congurations. One iteration of irace
comprises the process of (i) generation of candidate congurations, (ii) execution
of the racing procedure, and (iii) update of the probabilistic model.
The irace package [17,18] is an implementation of irace that is publicly avail-
able as an R package. Recently, version 2.0 of the software was released, which
implements an elitist racing procedure [17]. Dierently from the non-elitist rac-
ing procedure on which the rst version of irace is based, elitist irace evaluates
congurations on a set of problem instances that increases in size in every iter-
ation of irace. In particular, in elitist irace, an elite conguration carried over
from the previous iteration cannot be eliminated until a better conguration is
evaluated on the same instances as the elite one, including all instances on which
the elite conguration was previously evaluated and at least one new instance.
In more detail, elitist irace works as follows (see Fig. 1). In the rst iteration,
congurations are sampled uniformly at random from the given conguration
space. These congurations are evaluated on T first instances, after which the
Fig. 1. Illustration of the 1st and 2nd iteration of a run of irace using T first = 3,
T each = 1 and T new = 1.
rst statistical test is applied, and the congurations that are signicantly worse
performing than the best ones are eliminated. This elimination test is performed
every T each instances until the termination criterion of the iteration is met. The
surviving congurations at the end of the iteration (elite congurations) are used
to update a probabilistic model from which new congurations are sampled.
The set of congurations evaluated in the next iteration is composed of the elite
congurations and newly sampled congurations (non-elite ones). Algorithm 1
shows the pseudo-code of the race performed at each iteration of irace. Instances
are evaluated following an execution order that is built interleaving new and old
instances (procedure generateInstancesList in line 1).
More precisely, the instance list includes T new previously unseen instances,
followed by the list of previously evaluated instances (I old ), and nally, enough
new instances to complete the race. I old is randomly shued to avoid a bias
that could result from always using the same instance order. A race may termi-
nate even before evaluating all instances in I old (e.g. when a minimum number
of congurations is reached), and, as a result, each elite conguration may be
evaluated on some instances in I old . When the race nishes, irace therefore mem-
orises which conguration has been evaluated on which instance. (Line 7 updates
the elite status, and line 10 tests this condition.) In line 6, the congurations
are evaluated on instance I[i] with a maximum execution time of bmax . If an
elite conguration was already previously evaluated on I[i] (i.e., I[i] I old ), its
result on that instance is read from memory. When the statistical elimination
test is applied, only non-elite congurations (new ) may be eliminated and elite
ones are kept until they become non-elite. A conguration becomes non-elite if
all instances in I old on which it has previously been evaluated have been seen in
a race. Finally, the race returns the best congurations found, which will become
elite in the next iteration. For more details about irace, see [17].
3 ParamILS and Adaptive Capping
ParamILS [13] is an iterated local search [19] procedure that searches in a para-
meter space dened by categorical parameters only; for conguring numerical
parameters with ParamILS, these need to be discretised. ParamILS uses a rst-
improvement local search algorithm that explores, in random order, the one-
exchange neighbourhood of the current conguration.
There are two versions of ParamILS, BasicILS and FocusedILS, which dier
in the number of instances evaluated when comparing two congurations [13].
BasicILS compares congurations by evaluating them on a xed number N of
instances, while FocusedILS varies the number of instances according to the
quality of the congurations to be tested. The number of instances used in
the comparison is adjusted based on the dominance criterion, by which a
conguration j is dominated by a conguration i if (1) i has been evaluated
in at least as many instances as j and (2) the aggregated performance of i is
better or equal than the one of j on the Nj instances on which j has been
evaluated. When no dominance can be established between two congurations,
Algorithm 1. Racing procedure in elitist irace

Inputs are a set of newly generated configurations ( new ), a user-provided maximum execution
time (bmax ), the number of new initial instances (T new ), the list of unseen instances (I new ),
a set of elite configurations ( elite ), the list of instances on which elite were previously
evaluated (I old ), and a Boolean predicate isElite(, I) that returns true if configuration
elite was previously evaluated on instance I I old .In the first iteration of irace, elite and
I old are empty and all entries of isElite(, ) are set to false.
Input: elite , new , bmax , T new , I old , I new , isElite(, )

Output: Best configurations found in the race.
begin
1 I generateInstancesList (T new , I old , I new )
2 i1
3 i new elite
4 while termination() do
# execute elites only when needed; I[i] is the ith entry of instance list I
5 exe i \ { elite | isElite(, I[i])}
6 execute ( exe , bmax , I[i])
7 isElite(, I[i]) false elite
8 if mustTest(i) then
9 i+1 eliminationTest( i , {I[1], . . . , I[i]})
# keep configurations that are still elite

10 i+1 i+1 { elite | II old isElite(, I)}
11 else
12 i+1 i
13 ii+1
14 return i
the number of instances seen by the conguration with less instances evaluated
is increased until both congurations have seen the same number of evaluations.
The execution of a conguration on each instance is always bounded by a dened
maximum execution time (cut-o time).
The adaptive capping technique further bounds the execution of a cong-
uration by using the running time of good congurations as a bound in running
time that is often less than the user-specied cut-o time. Using this technique
can signicantly reduce the time wasted in the evaluation of poor performing
congurations. Adaptive capping adjusts the bound on running time according
to the number of instances to be used in the comparison, and for this reason, it
can be sensitive to the ordering of the given instances. There are two types of
adaptive capping: trajectory preserving and aggressive capping [13]. The rst of
these bounds the running time of new congurations using the performance of
the currently best conguration of each ParamILS iteration as reference, while
the second additionally uses the performance of the overall best conguration
multiplied by a factor, set to two by default, for bounding. This factor controls
the aggressiveness of the capping strategy. Further details on adaptive capping
can be found in [13].
Algorithm 2. Racing procedure in iracecap

For the description of the inputs, see Algorithm 1.
Input: elite , new , bmax , T new , I old , I new , isElite(, )

Output: Best configuration set found in the race.
begin
1 I generateInstancesList (T new , I old , I new )
2 execute ( elite , bmax , {I[1], . . . , I[T new ]})
3 i1
4 i new elite
5 while termination() do
6 bi calculateEliteBound ( elite , {I[1], . . . , I[i]})
7 exe i \ { elite | isElite(, I[i])}
8 execute ( exe , bi , I[i])
9 isElite(, I[i]) false elite
# dominancecriterion elimination
10 i+1 eliminationDominance ( i , {I1 , . . . , I[i]})
# statistical test elimination
11 if mustTest (i) then
12 i+1 eliminationTest ( i+1 , {I1 , . . . , I[i]})
# keep configurations that are still elite

13 i+1 i+1 { elite | II old isElite(, I)}
14 i i+1
15 return i
4 Adaptive Capping in irace

In this section, we describe a new version of irace that adopts the ideas underlying
adaptive capping in the racing procedure. This new version, iracecap , introduces
two new components to the algorithm: (1) the adaptive running time bound,
used to limit the running time of new congurations on previously seen and
initial instances, and (2) dominance elimination, a procedure that discards
poorly performing congurations. Algorithm 2 shows the outline of the racing
procedure implemented in iracecap ; it follows the same structure as elitist irace,
described in Sect. 2. The elite congurations are rst run on the set of initial
instances before the start of the race (Line 2). Line 6 calculates an initial running
time bound based on the running times of the elite congurations. Let pji be
the average computation time of a conguration j up to instance I[i] in the
current iteration. Then, the bound bi for running new congurations on instance
I[i] is equal to medianj elite {pji }. (Median is chosen to be consistent with the
elimination based on dominance described next.) The bound bi can be computed
only for previously evaluated instances (including the initial instances); for any
other instance, we set the bound to the cut-o time, bmax . This running time
bound provides a reference of the minimum performance new congurations
should obtain in order to compete with the current elite congurations.
The maximum running time kij for each conguration j on instance I[i] is
computed by procedure execute in line 8 using the value of bi as follows:

kij = bi i + bmin pji1 (i 1) (1)

b
max
if kij > bmax ,

kij = min{bi , bmax } if kij 0, (2)

j
ki otherwise;
where bmin is a constant that represents a minimally measurable running time
dierent from zero (set to a default value of 0.01). Intuitively, kij is the time
remaining for a conguration j to improve over the median elite conguration.
We implemented a dominance-based elimination procedure inspired by the
domination criterion described in Sect. 3. We compare the median performance
of the elite congurations set (elite ) on the list of instances {I[1], . . . , I[i]}
considered so far with the performance of the new congurations as follows:
Medians elite {psi } + bmin < pji (3)
where psi is the mean running time of conguration s on instances

{I[1], . . . , I[i]}, and bmin is the constant dened in Eq. (2). Other choices than
the median are possible and may be considered in future work. We eliminate
congurations as soon as they become dominated, that is, the dominance-based
elimination is applied after every instance seen within an iteration of irace.
5 Experiments
In this section, we study the impact of introducing the previously described cap-
ping procedure into irace. We compare the performance of the nal congurations
obtained by elitist irace and iracecap using dierent settings.
In our performance assessments of iracecap , we use ve conguration scenarios

taken from previous experimental studies of other automatic algorithm con-
guration methods, in particular, ParamILS and SMAC. These scenarios use
CPLEX [16], Lingeling [5] and Spear [3] as target algorithms, and involve para-
meter spaces with 74, 137 and 26 parameters, respectively. Their principal char-
acteristics are as follows:
CPLEX - Regions100 [12,13]. 5 s cut-o time, 18 000 s total conguration

budget, and a training and testing set of 1000 mixed integer programming
(MIP) instances each. The instances encode a combinatorial auction winner
determination problem with 100 goods and 500 bids.
CPLEX - Regions200 [11,13]. 300 s cut-o time, 172 800 s total conguration
budget, and a training and testing set of 1000 MIP instances each.These
instances are encodings of a combinatorial auction winner determination
problem with 200 goods and 1000 bids.
CPLEX - Corlat [11]. 300 s cut-o time, 172 800 s total conguration budget,
and a training and testing set of 1000 MIP instances each.
Lingeling [14]. 300 s cut-o time, 172 800 s total conguration budget, and a
training and testing set of 299 and 302 SAT instances, respectively. These
instances were obtained from the 2014 Congurable SAT Solver Competition
(CSSC) [14].
Spear [13]. 300 s cut-o time, 172 800 s total conguration budget, and a train-
ing and testing set of 302 SAT-encoded software verication instances each.
The instance les for these scenarios are also available from the Algorithm
Conguration Library (AClib) [15]. AClib species a cut-o time of 10 000 s for
the CPLEX scenarios, which stems from their initial use in conjunction with the
CPLEX auto-tuning tool. Following the experiments in [11, Sect. 5]), we use a
cut-o time of 300 s.1 Another minor dierence is that we usedversion 12.4 of
CPLEX, which was installed on our system, while AClib proposes to use version
12.6. However, there is no obvious reason to suspect that the particular version
of CPLEX should aect our conclusions on the eect of capping inside irace, and
we do not directly compare to results for the original AClib scenarios. Moreover,
although both irace and SMAC are able to handle non-discrete parameter spaces,
for ParamILS, all parameters have to be discretised, with all possible values
specied explicitly in the scenario denition. There is some evidence that the use
of non-discrete parameter spaces, where possible, leads to improved results [12],
thus giving an advantage to both irace and SMAC over ParamILS, unrelated
to the capping mechanism, which is the focus of our comparison presented in
Sect. 6. To avoid this bias, we only consider the variants of the scenarios where
all parameters are discretised and explicitly specied.
In all our experiments, we used the t-test to eliminate congurations within
irace, as previously recommended for running time minimisation [21]. The com-
parisons presented in the following are based on 20 independent runs of all
conguration procedures; multiple independent congurator runs are performed
due to the inherent randomness of the conguration procedures and the con-
guration scenarios. The experiments were run on one core of a dual-processor
2.1 GHz AMD Opteron system with 16 cores per CPU, 16 MB cache and 64 GB
RAM, running Cluster Rocks 6.2, which is based on CentOS 6.2.
In our empirical analysis of iracecap , we use mean running time as the per-
formance criterion to be optimised by irace. Runs that time out due to reaching
the cut-o time are then counted at this maximum cut-o time. In the litera-
ture, unsuccessful runs are often more strongly penalised, computing eectively
the number of timed out runs multiplied by a penalty factor pf plus the mean
computation time of the successfully terminated runs. In fact, the penalty fac-
tor pf converts the bi-objective problem of minimising the number of timed-out
runs and mean time of successful runs into a single-objective problem. In this
1
A higher cut-o time, as used in AClib, would be detrimental for conguration pro-
cedures such as iracecap , as time-outs would very strongly impact the number of con-
gurations that can be evaluated. On the other hand, there are various techniques,
such as early termination of ongoing runs or the initial use of smaller maximum
cut-o times, to address this problem. In the literature, the use of smaller cut-o
times has been suggested as a possible remedy [12, footnote 9].
section, runs of irace attempt to minimise mean running time (with pf = 1),
and we therefore assessed the performance of the resulting target algorithm
congurations using this performance metric. In the supplementary material,
we additionally present results for evaluating congurations using pf = 10 and
pf = 100. In the literature, pf = 10 is commonly used and referred to as PAR10;
consequently, in Sect. 6, all congurator runs and target algorithm evaluations
are performed using PAR10 scores.
5.2 Experimental Results
We rst compare the results obtained by elitist irace and iracecap , using their
respective default settings. Table 1 presents performance statistics over the 20
runs of both irace versions. The implementation of the proposed capping proce-
dure proves to be benecial for the scenarios used in these experiments. For the
Regions 100, Regions 200, Corlat, and Spear scenarios, the results obtained by
iracecap are signicantly better than those of elitist irace, while for the Lingeling
scenario, the results are not signicantly dierent (however, iracecap still achieves
a better mean than irace).
Table 1. Summary statistics of the distribution of observed mean running time and
percentage of timed out evaluations of 20 runs of iracecap and elitist irace (irace) on
test sets for the various conguration scenarios. We show the rst and second quartile
(q25 and q75, respectively), the median, the mean, the standard deviation (sd) and the
variation coecient (sd/mean). Wilcoxon test p-values are reported in the last line.
Statistically signicantly better results (at = 0.05) are indicated in bold-face and
lowest mean running times in italics.
Regions 100 Regions 200 Corlat Lingeling Spear

iracecap irace iracecap irace iracecap irace iracecap irace iracecap irace
%timeout 0.08 0.085 0.01 0.015 0.695 1.205 8.377 8.659 0.397 3.328
q25 0.327 0.374 9.487 10.983 8.616 13.526 42.379 44.274 3.028 4.776
mean 0.338 0.395 10.498 13.231 11.899 15.935 45.501 46.923 4.116 13.068
median 0.332 0.401 10.469 12.871 9.688 14.911 44.453 47.034 3.765 14.617
q75 0.34 0.413 10.75 14.256 13.941 18.436 48.996 49.738 4.242 19.993
sd 0.018 0.033 1.335 2.908 5.645 4.325 3.799 3.658 1.848 8.092
sd/mean 0.054 0.082 0.127 0.22 0.474 0.271 0.083 0.078 0.449 0.619
p-value 5.7e-06 0.0001049 0.0055809 0.2942524 0.0002613
The elimination criterion of the capping procedure in iracecap only considers

aggregated running time rather than its statistical distribution, and it is not
obvious whether this renders the criterion always stricter than the statistical
test at the core of irace, which could, in principle, render the latter superuous.
Figure 2 shows the mean percentage of live congurations selected to be elim-
inated by the capping procedure and the statistical test (lines), and the mean
percentage of initial congurations that become elite congurations at the end
of the iteration (bars). (For results on all other scenarios, see Figure A.2.) The
capping procedure selects more congurations for elimination than the statistical
test in all stages of the search, while the statistical test is mainly able to eliminate
congurations in the initial phases of the search. As the race progresses, capping
elimination quickly becomes mainly responsible for eliminating congurations,
illustrating the importance of introducing it into irace.
Fig. 2. Mean percentage of congurations selected for elimination by the capping proce-
dure and the statistical test (solid and dashed lines respectively), and mean percentage
of initial congurations that become elite congurations at the end of the iteration
(bars). Means obtained across 20 independent runs of iracecap on the Regions 200 and
Spear scenarios.
The capping mechanism of iracecap and the increased elimination of congu-

rations induce a highly intensied search. On average, iracecap performs in part
many more iterations than irace and shows a lower average number of elite con-
gurations per iteration. This results in an increased number of congurations
sampled overall and instances used for evaluation (Table 2).
Table 2. Statistics over 20 independent runs of iracecap and irace: mean number of
iterations performed (iterations), mean number of instances used in the evaluation
(instances), mean overall sampled congurations (candidates), mean elite congura-
tions per iteration (elites) and mean total executions (executions).
mean Regions 100 Regions 200 Corlat Lingeling Spear

iracecap irace iracecap irace iracecap irace iracecap irace iracecap irace
iterations 253.5 28.3 85.8 17.1 68.7 13 27.4 10.5 67.0 7.6
instances 258.6 47.6 91.1 36.7 75.1 29.4 35.5 26.3 83.2 16.3
candidates 27914 1136 5191 285 5318 242 2595 214 11193 718
elites 1.09 6.88 1.25 7.12 1.82 7.80 3.28 8.90 2.26 5.99
executions 30604 8362 6779 2770 8873 3147 5218 2878 28039 6109
5.3 Additional Analysis of iracecap
In what follows, we examine in more detail the impact of some specic parameter
settings of iracecap on its performance. For the sake of conciseness, we will only
discuss overall trends;detailed results are found in supplementary material [20].
Instance order. The order of the instances may introduce a bias in irace when
the conguration scenario involves a heterogeneous instance set. By default, irace
shues the order of the training instances. Without this shuing, irace evalu-
ates the instances in the order provided by the user. Since the set of previously
used instances is evaluated in every iteration, elitist irace randomly permutes the
order of previously seen instances (I old ) before each iteration to further avoid
any bias that the previous order may introduce. Table A.2 compares the results
obtained by iracecap with and without this instance reshuing. For most bench-
mark scenarios, disabling instance reshuing produces better mean results and
fewer timed-out runs; for Regions 200 and Lingeling, these dierences are sta-
tistically signicant. The main exception is the Spear scenario, where reshuing
leads to much improved results; this is probably due to the fact that this scenario
contains a very heterogeneous instances set.
These results suggest that the impact of reshuing depends on the given
conguration scenario; we conjecture that for more heterogeneous instance sets,
reshuing the instance set becomes increasingly important. Investigating this
conjecture in detail is an interesting direction for future work.
Confidence level of statistical test. The dominance criterion eliminates more
congurations than the statistical test. Lowering the condence level of the sta-
tistical test should lead to an even higher elimination rate of the latter and
possibly improve the ecacy of the overall conguration process. We explored
this possibility by lowering the condence level in iracecap from its default value
of 0.95 to 0.75. Table A.3 in the supplementary material shows the impact of
this change. The eects on the elimination of congurations can be observed
in Figure A.5 in the supplementary material. As expected, the statistical test
eliminates more congurations when setting the condence level to 0.75. This
also results in a small increase in the overall number of congurations evaluated
and a reduction of the mean number of elite congurations (see Table A.4 in the
supplementary material). The more aggressive test slightly improved the perfor-
mance for three scenarios, yielding signicantly better results for Regions 200. In
contrast, a condence level of 0.75 results in slightly worse performance on the
Spear scenario, indicating that the eliminations performed with lower condence
can be premature.
If we completely disable statistical testing (condence level 1.0), the perfor-
mance of iracecap improves on Regions 100 and Regions 200, as seen in Table A.5
in the supplementary material. This suggests that the statistical test can prema-
turely eliminate congurations based on an incorrect criterion. Despite this, we
still recommend keeping the default condence level of 0.95, as a safe-guard that
may be useful for conguration scenarios with possibly very dierent properties
from the ones we are testing here.
Log-transformation of running times. When used for running time min-

imisation, irace makes use of the t-test for elimination. However, the potentially
very large variability of target algorithm running times [10] often renders the
distribution of running times far from normal, a situation that may be allevi-
ated by using a transformation of running times in particular, a logarithmic
transformation. Applying this transformation has, however, only a signicant
eect on the Regions 200 scenario. Increasing the dierence between the per-
formance of congurations makes the elimination more aggressive and, as seen
in other experiments, the Regions 200 scenario benets greatly of this increased
intensication. For the other scenarios, the impact on performance is negligible,
as seen in Table A.6 in the supplementary material, probably due to the minor
impact of the statistical test on the elimination of congurations.
Number of initial new instances. Finally, we performed experiments to eval-
uate the impact of adding new instances at the beginning of each race. If no new
instances are added at the start, then new congurations can only become elite
by performing better on exactly the same instances on which the current elites
performed well in previous races. Even though the new congurations may be
better on instances not seen yet, they may be eliminated before seeing them,
unless those new instances are evaluated at the start. On the other hand, there
are no running times available for new instances; this issue is addressed by rst
running the elite congurations on the new instances to avoid wasting too much
computation time on possibly poor newly sampled candidate congurations.
Table A.7 in the supplementary material shows the results for setting the num-
ber of new initial instances (T new ) to 0 (new instances are never added at the
start of each race), 1 (the default setting) and 5. As previously observed for
elitist irace [17], a larger value of T new improves the performance of iracecap for
the Spear scenario. While for the other scenarios, the dierences are minor, there
appears to be a tendency for the default value of 1 (or perhaps even a slightly
larger value, such as 2 or 3) to result in the most robust behaviour.2
6 Comparison to Other Configurators
We compare the results obtained by iracecap with two other automatic congu-
rators available in the literature, ParamILS and SMAC. Both have been widely
used in the literature for running time minimisation. SMAC and ParamILS, as
well as irace, were run using default settings. We chose not to include instance
features in the conguration process and use only fully discretised conguration
spaces; this was done to isolate as much as possible the impact of the new cap-
ping mechanism in irace, and to examine whether it would become competitive
2
Setting T new to 0 may be benecial for scenarios with a very large cut-o time,
as used by default in AClib for the CPLEX scenarios. This should help to aggres-
sively bound the running time at the start of each race, by using the running times
of the elite congurations, thus avoiding the high cost of evaluating possibly poor
congurations with a very large cut-o time.
with other congurators that already used this technique. Considering features
or non-discrete parameter spaces would introduce additional factors that are
likely to aect performance beyond the impact of capping. Nevertheless, SMAC
can also use instance features in the conguration process, which may improve
its results; therefore, the results obtained here should be considered with caution
for those scenarios in cases where these features are available. Yet, identifying
how much of the improvement is due to instance features or due to dierences
in the capping methods between SMAC and other congurators would require
a more extensive analysis that is left for future research. Additionally, SMAC
and irace can handle real-valued parameters and, as already shown for SMAC in
[12], doing so may further improve performance.
As mentioned previously, we ran iracecap , SMAC and ParamILS using the
PAR10 evaluation on the scenarios described in Sect. 5. Table 3 shows the mean
PAR10 execution times obtained from 20 runs of the congurators. In the on-
line supplementary material, we present results with other penalty factors from
{1, 10, 100}. The table shows the p-values obtained from the Wilcoxon signed-
rank test comparing the performance of the two congurators with the lowest
mean PAR10 score. iracecap obtains the statistically signicantly lowest mean
on the Regions 200, Corlat, and Lingeling scenarios, while SMAC obtains the
statistically signicantly lowest mean on the Spear scenario. On the Regions
100 scenario, iracecap obtains the lowest mean performance value, though its
performance is not statistically dierent from that of ParamILS.
It is known that trajectory-based local search methods, such as ParamILS,
can exhibit high performance variability over multiple independent runs due
to search stagnation. A common practice for dealing with this situation, and
for reducing the overall wall-clock time of the conguration process by means
Table 3. Statistics over the mean PAR10 performance and percentage of timed-out
instances from 20 runs of iracecap , SMAC and ParamILS. Wilcoxon test p-values (sig-
nicance 0.05). Signicantly better results in bold and best mean in cursive.
q25 mean median q75 sd sd/mean %timeout

Regions 100 p-value: ParamILS 0.318 0.38 0.37 0.416 0.066 0.173 0.130
0.5958195 SMAC 0.45 0.478 0.473 0.499 0.055 0.116 0.045
iracecap 0.32 0.372 0.365 0.395 0.057 0.154 0.095
Regions 200 p-value: ParamILS 9.412 11.656 10.359 13.606 3.348 0.287 0.005
0.03276825 SMAC 14.205 17.917 16.452 21.925 5.419 0.302 0.045
iracecap 8.854 9.926 9.349 10.533 1.459 0.147 0.005
Corlat p-value: ParamILS 30.924 193.303 48.772 74.309 360.42 1.865 5.945
0.0083084 SMAC 30.866 45.847 39.855 63.805 20.903 0.456 1.005
iracecap 12.24 27.974 26.763 33.902 19.426 0.694 0.62
Lingeling p-value: ParamILS 250.115 292.529 298.942 327.679 51.497 0.176 9.023
0.0362339 SMAC 266.792 283.907 289.153 298.64 25.121 0.088 8.758
iracecap 244.313 263.651 259.768 271.119 31.736 0.12 8.113
Spear p-value: 9.5e-06 ParamILS 3.037 88.083 12.094 40.877 188.929 2.145 2.815
SMAC 1.6 3.416 1.746 2.511 3.733 1.093 0.05
iracecap 5.666 23.741 22.3 25.872 21.512 0.906 0.662
Table 4. Statistics over the mean PAR10 performance for the best-out-of-ten runs
sampled from the 20 original runs of iracecap , SMAC and ParamILS. Wilcoxon test
p-values ( = 0.05). Signicantly better results are shown in bold-face and best mean
values in italics.
q25 mean median q75 sd sd/mean

Regions 100 p-value: 8.83e-05 ParamILS 0.303 0.305 0.303 0.307 0.003 0.011
SMAC 0.386 0.392 0.386 0.391 0.012 0.031
iracecap 0.312 0.314 0.314 0.314 0.002 0.007
Regions 200p-value: 0.0001417 ParamILS 8.589 8.871 8.999 9.139 0.266 0.03
SMAC 9.82 11.2 10.106 12.989 1.784 0.159
iracecap 8.456 8.523 8.518 8.581 0.069 0.008
Corlatp-value: 0.0010241 ParamILS 8.284 10.58 9.58 9.58 4.163 0.393
SMAC 17.801 19.898 19.832 19.832 3.016 0.152
iracecap 7.959 8.194 7.959 8.256 0.627 0.076
Lingelingp-value: 0.6341078 ParamILS 210.753 218.874 210.753 223.379 13.256 0.061
SMAC 230.175 241.659 236.26 257.77 12.85 0.053
iracecap 220.093 220.212 220.093 220.212 0.212 0.001
Spearp-value: 9e-05 ParamILS 1.911 2.075 2.012 2.073 0.27 0.13
SMAC 1.454 1.462 1.454 1.463 0.018 0.013
iracecap 2.154 2.497 2.184 2.497 0.581 0.233
of parallelisation, is to perform multiple independent congurator runs concur-

rently and to return the best conguration found in any of these. This may not
always be feasible when the average running times of congurations on instances
are high, e.g., in the range of hours, in which case the parallelisation features of
irace would be very useful. Nevertheless, we mimic this commonly applied app-
roach and compare the performance of ParamILS, SMAC and iracecap based on
the following resampling approach: From the 20 values of mean PAR10 perfor-
mance previously obtained for each congurator on the test set of each scenario,
we sample 10 values (uniformly at random and without repetition) and take the
best of these samples. This is equivalent to running the congurator 10 times
and determining the best of the congurations thus obtained. We repeat this
process 20 times to obtain 20 replicates of the experiment. Table 4 shows the
results thus obtained. ParamILS benets most from multiple independent runs,
achieving the statistically signicantly best performance on the Regions 100
scenario and the best mean performance (though not statistically signicantly
dierent from that of iracecap ) on Lingeling. iracecap produces the statistically
signicantly best results on Regions 200 and Corlat, while SMAC shows the
best mean performance for the Spear scenario.
7 Conclusions
In this work, we have extended irace, an automatic algorithm conguration

procedure primarily designed for solution quality optimisation, with an adap-
tive capping mechanism. We have demonstrated that this results in substantial
improvements in the ecacy of irace for running time minimisation, and our new
iracecap congurator reaches state-of-the-art performance on prominent congu-
ration scenarios. This considerably broadens the range of conguration scenarios
on which irace should be seen as one of the methods of choice.
In future work, it would be interesting to explore which characteristics of
a conguration scenario makes it particularly amenable to dierent variants of
adaptive capping. Furthermore, we would like to investigate under which cir-
cumstances iracecap performs better (or worse) than other state-of-the-art con-
gurators, notably SMAC [12], ParamILS [13] and GGA++ [1]. We see this as
an important step towards automatic selection of the congurator expected to
perform best on a given scenario. This could improve the state of the art in auto-
matic algorithm conguration and further boost the appeal of the programming
by optimisation (PbO) software design paradigm [9], which crucially depends on
maximally eective congurators.
Acknowledgments. This research was supported through funding through COMEX

project (P7/36) within the Interuniversity Attraction Poles Programme of the Bel-
gian Science Policy Oce. Thomas Stutzle acknowledges support from the Belgian
F.R.S.-FNRS, of which he is a Senior Research Associate. Holger Hoos acknowledges
support through an NSERC Discovery Grant.
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Duality Gap Analysis of Weak Relaxed
Greedy Algorithms
Sergei P. Sidorov1(B) and Sergei V. Mironov2

1
Mechanics and Mathematics Department, Saratov State University,
Saratov, Russian Federation
SidorovSP@info.sgu.ru
2
Computer Science and Information Technologies Department,
Saratov State University, Saratov, Russian Federation
MironovSV@info.sgu.ru
Abstract. Many problems in machine learning can be presented in the

form of convex optimization problems with objective function as a loss
function. The paper examines two weak relaxed greedy algorithms for
nding the solutions of convex optimization problems over convex hulls
of atomic sets. Such problems arise as the natural convex relaxations
of cardinality-type constrained problems, many of which are well-known
to be NP-hard. Both algorithms utilize one atom from a dictionary per
iteration, and therefore, guarantee designed sparsity of the approximate
solutions. Algorithms employ the so called gradient greedy step that
maximizes a linear functional which uses gradient information of the ele-
ment obtained in the previous iteration. Both algorithms are weak in
the sense that they solve the linear subproblems at the gradient greedy
step only approximately. Moreover, the second algorithm employs an
approximate solution at the line-search step. Following ideas of [5] we
put up the notion of the duality gap, the values of which are computed
at the gradient greedy step of the algorithms on each iteration, and
therefore, they are inherent upper bounds for primal errors, i.e. dier-
ences between values of objective function at current and optimal points
on each step. We obtain dual convergence estimates for the weak relaxed
greedy algorithms.
Keywords: Greedy algorithms Convex optimization Sparsity

Duality gap
1 Introduction
Let X be a Banach space with norm . Let E be a convex function dened
on X. The problem of convex optimization is to nd an approximate solution to
the problem
E(x) min . (1)
xX
Many problems in machine learning can be reduced to the problem (1) with
E as a loss function [1]. In many real applications it is required that the optimal
https://doi.org/10.1007/978-3-319-69404-7_18
252 S.P. Sidorov and S.V. Mironov
solution x of (1) should have a simple structure, e.g. be a finite linear combi-
nation of elements from a dictionary D in X. In another words, x should be a
sparse element with respect to the dictionary D in X. Of course, one can sub-
stitute the requirement of sparsity by a constraint on cardinality (i.e. the limit
on the number of elements used in linear combinations of elements from the
dictionary D to construct a solution of the problem (1)). However, it many cases
the optimization problems with cardinality-type constraint are NP-complete.
By this reason, practitioners and researchers in real applications choose to use
greedy methods. By its design, greedy algorithms is capable of producing sparse
solutions.
A set of elements D from the space X is called a dictionary (see, e.g. [15]) if
each element g D has norm bounded by one, g 1, and the closure of span
D is X, i.e. span D = X. A dictionary D is called symmetric if g D for every
g D. In this paper we assume that the dictionary D is symmetric.
As it was pointed out, practitioners and researchers would like to nd the
solutions of the optimization problem (1), which are sparse with respect to the
dictionary D, i.e. they are looking for solving the following problem:
E(x) inf , (2)

xm (D)
where m (D) is the set of all m-term polynomials with respect to D:

m

m (D) = xX : x= ci gi , gi D . (3)
i=1
One of the apparent choices among constructive methods for nding the best
m-term approximations are greedy algorithms. The design of greedy algorithms
allows us to obtain sparse solutions with respect to D. Perhaps, the Frank-Wolfe
method [2], which is also known as the conditional gradient method [3], is one
of the most prominent algorithms for nding optimal solutions of constrained
convex optimization problems. Important contributions to the development of
Frank-Wolfe type algorithms can be found in [46]. The paper [5] provides general
primal-dual convergence results for Frank-Wolfe-type algorithms by extending
the duality concept presented in the work [4]. Recent convergence results for
greedy algorithms one can nd in the works [714,1618,20].
This paper examines two weak relaxed greedy algorithms
Weak Relaxed Greedy Algorithm (WRGA(co)),
Weak Relaxed Greedy Algorithm with Error (WRGA())
for nding solutions of convex optimization problem, which are sparse with
respect to some dictionary, in Banach spaces. Primal convergence results for
the weak relaxed greedy algorithms were obtained in [15,18]. In this paper,
extending the ideas of [4,5] we force into application the notion of the duality
gap for weak relaxed greedy algorithms to obtain dual convergence estimates
for sparse-constrained convex optimization problems of type (2). In contrast to
Duality Gap Analysis of Weak Relaxed Greedy Algorithms 253
papers [15,18,19], in this paper we focus on obtaining dual convergence results

based on duality gap analysis.
It should be noted, that the paper of Temlyakov [15] shows that the greedy
algorithms (WRGA(co) and WRGA()) for nding the solutions of (2) with
respect to the dictionary D solve the problem (1) as well. In many real applica-
tions the dimension of the search space while is nite, but is too large. Therefore,
our interest lies in obtaining estimates on the rate of convergence not depending
on the dimension of X. Obviously, results for the innite Banach spaces provide
such estimates on the convergence rate. Following [15], we examine the problem
in an innite dimensional Banach space setting.
Note that duality for global convex minimization problems has been well-
examined. The recent paper [21] shows that not all nonconvex global minimiza-
tion problems are NP-hard and that the complexity of a problem depends essen-
tially on modeling and intrinsic symmetry of the problem.
2 Weak Relaxed Greedy Algorithms

We will suppose that function E is Frechet dierentiable. We note that it follows
from convexity of E that for any x, y
E(y) E(x) + E (x), y x,
where E (x) denotes Frechet dierential of E at x.

To solve the optimization problem (2), the paper [15] uses iterative search
optimizer described in Algorithm 1. The algorithm for each m 1 nds the next
element Gm by means of induction with use of
the current element Gm1
element m obtained in the gradient greedy step.
In the gradient greedy step we maximize a functional which uses gradient
information at element Gm1 of X obtained in the previous iteration of the
algorithm. Algorithm 1 belongs to the class of Frank-Wolfe type methods, since
at each current element Gm1 of X it drifts to a minimizer of the linearization
of the objective function E taken over the feasible set, which in this case is the
dictionary D.
We would like to note that the gradient greedy step of WRGA(co) is looking
for supremum over the dictionary D (not its convex hull A1 (D)), since points
from A1 (D) are mostly linear combinations of innite number of the dictionary
elements. Thus, the optimal solution obtained by this way is not obliged to be
sparse with respect to D.
Let := {tm } m=1 , tm [0, 1], be a weakness sequence. To solve the sub-
problem supsD E (Gm1 ), s Gm1 exactly may be too expensive in many
real cases. Algorithm 1 uses the weakness sequence in the gradient greedy
step to nd approximate minimizer m instead, which has approximation (mul-
tiplicative) quality at least tm in step m. That is why the algorithm is called
weak.
Algorithm 1. Weak Relaxed Greedy Algorithm (WRGA(co))

begin
Let G0 = 0;
for each m = 1, 2, . . . , M do
(Gradient greedy step) Find the element m D such that
E (Gm1 ), m Gm1 tm sup E (Gm1 ), s Gm1 ;
sD
(Line-search step) Find the real number 0 m 1, such that
E ((1 m )Gm1 + m m ) = inf E ((1 )Gm1 + m ) ;
01
(Update step) Gm = (1 m )Gm1 + m m ;
end
The line-search step of Algorithm 1 nds the best point lying on the line
segment between the current point Gm1 and m .
Let := {x X : E(x) E(0)} and suppose that is bounded. As it
turns out, the convergence analysis of greedy algorithms essentially depends on
a measure of non-linearity of the objective function E over set , which can
be depicted via the modulus of smoothness of function E.
Let us remind that the modulus of smoothness of function E on the bounded
set can be dened as
1
(E, u) = sup |E(x + uy) + E(x uy) 2E(x)|, u > 0. (4)
2 x,y=1
E is called uniformly smooth function on if limu0 (E, u)/u = 0.

Let A1 (D) denote the closure (in X) of the convex hull of D.
Exploiting the geometric properties of the objective function E, the paper [15]
proves the following estimate of the convergence rate of the WRGA(co).
Theorem 1. Let E be a uniformly smooth convex function with modulus of

smoothness (E, u) uq , 1 < q 2, > 0. Then, for a weakness sequence
= {tk }
k=1 , 0 < tk 1, k = 1, 2, . . ., we have for any f A1 (D) that
m
1q
q
E(Gm ) E(f ) 1 + C1 (q, ) tpk , p := , m 2, (5)
q1
k=1
where C1 (q, ) is positive constants not depending on k.
The paper [18] notes that values of E may not be calculated exactly for many
real application problems. Moreover, very often the exact optimal value of in
the problem
inf E ((1 )Gm1 + ) (6)
01
in Step 2 of WRGA(co) can not be found. Therefore, the paper [18] examines
the weak relaxed greedy algorithm with error (Algorithm 2), which is a slightly
modied version of WRGA(co) with changing in the second step. In comparison

with the WRGA(co), WGAFR() uses the error in the line-search step with
the aim of getting the optimization problem of a dierent kind. It may have
better complexity than the original optimization problem (6).
Algorithm 2. Weak Relaxed Greedy Algorithm with Error

(WRGA())
begin
Let > 0 and G0 = 0;
for each m = 1, 2, . . . , M do
(Gradient greedy step) Find the element m D such that
E (Gm1 ), m Gm1 tm sup E (Gm1 ), s Gm1 ;
sD
(Line-search step) Find the real number 0 m 1, such that
E ((1 m )Gm1 + m m ) inf E ((1 )Gm1 + m ) + ;
01
(Update step) Gm = (1 m )Gm1 + m m ;
end
The following estimate of the convergence rate of the WRGA() is proved in

the paper [18].
Theorem 2. Let E be uniformly smooth on A1 (D) whose modulus of smooth-
ness (E, u) satisfies (E, u) uq , 1 < q 2, > 0. If tk = , k = 1, 2, . . .,
then WRGA() satisfies
E(Gm ) E C(q, , , E)m1q , m 1/q ,
where E := inf E(x).
f A1 (D)
3 Dual Convergence Results

3.1 Duality Gap
Following ideas of [5], let us introduce the notion of the duality gap for opti-
mization problem as follows.
Definition 1. Let G A1 (D). Let us define the (surrogate) duality gap g(G) at
element G by
g(G) := supE (G), G s. (7)
sD
The decisive property of the duality gap is the following one.

Proposition 1. Let E be a uniformly smooth convex function defined on
Banach space X. Let x = arg minxA1 (D) E(x). For any G A1 (D)
E(G) E(x ) g(G).
Proof. Since E is convex on X we have for any y A1 (D)
E(y) E(x) + E (x), y x E(x) sup E (x), x s. (8)

sA1 (D)
Lemma 2.2 in [15] states that
sup E (x), x s = supE (x), x s. (9)

sA1 (D) sD
Then Proposition follows from (8) and (9) with y = x .
Proposition 1 shows that the duality gap g(G) is a bound for the current
approximation E(G) to the optimal solution E(x ).
The duality gap g is calculated as a derivative product on every iteration of
both Algorithms 1 and 2. If the linearized problem at the gradient greedy step
for element Gm1 has optimal solution m , then the element m is a reference
for the current duality gap
g(Gm1 ) = E (Gm1 ), Gm1 m .
Such references for the approximation quality of current iteration can be used
as a stopping criterion, or to verify the numerical stability of an optimizer.
Theorems 1 and 2 give upper estimates for primal errors for WRGA(co) and
WRGA(), respectively. In the next subsections we will obtain dual estimates
for the algorithms in terms of duality gap g.
3.2 Dual Convergence Result for WRGA(co)
We need some preliminary results.
Lemma 1. Let E be a uniformly smooth convex function defined on Banach

space X. Let (E, u) denote the modulus of smoothness of E. Then the following
inequality holds:
E(Gm ) E(Gm1 ) + inf (tm g(Gm1 ) + 2(E, 2)), m = 1, 2, . . . .

01
Proof. The denition of Gm in Step 3 of WRGA(co) implies that
Gm = (1 m )Gm1 + m m = Gm1 + m (m Gm1 ),
and
E(Gm ) = inf E(Gm1 + (m Gm1 )). (10)
01
It follows from Lemma 2.3 of [15] that

E(Gm1 + (m Gm1 ))
E(Gm1 ) E (Gm1 ), m Gm1 + 2(E, 2). (11)
The step 1 of WRGA(co) gives
E (Gm1 ), m Gm1
tm supE (Gm1 ), s Gm1 = tm g(Gm1 ). (12)
sD
Then the lemma follows from (10), (11) and (12).
Lemma 2. Let = {tk }

k=1 , < tk 1, k = 1, 2, . . ., for a fixed real > 0.
Denote
m
q
sm := sm (m, ) := tpk , p = , 1 < q 2. (13)
q1
k=1
Let 0 < < 1 be a real and M be an integer. Then

p 1q 1q
s1q
[M ]+1 ( ) sM ,
where square brackets denote the integer part.
Proof. Denote m0 := [M ] + 1. We have m M and

0
m0 p m 0 p
k=1 tk k=1 m0 p s[M ]+1
M p M = p , or p .
t
k=1 k k=1 1 M sM
Theorem 3. Let E be a uniformly smooth convex function defined on Banach

space X. Let (E, u) be the modulus of smoothness of E and suppose that
(E, u) uq , 1 < q 2. Let = {tk }k=1 , < tk 1, k = 1, 2, . . ., be a
weakness sequence, > 0. Assume that WRGA(co) is run for M > 2 iterations.
Then there is an iterate 1 m M such that
M
1q
q
g(Gm ) C2 tpk , p := (14)
q1
k=1
1q
where C2 := C2 (q, ) := (min{1, C1 (q, )}) and depends only on M, q, , .
Proof. It follows from Theorem 1 that for any f A1 (D)

m
1q
q
E(Gm ) E(f ) C2 tpk , p := , m 2. (15)
q1
k=1
Let us assume (by contradiction) that
g(Gm ) C2 s1q
M , (16)
for all [M ] + 1 m M (0 < < 1 is xed and will be chosen later), sM is

dened in (13).
It easy to check that tq1 s1q

m 1 for all m = 1, 2, . . .. Then Lemma 1 with
= tq1 s1q
m implies that
E(Gm+1 ) E(f ) E(Gm ) E(f ) tq1 s1q q 1q q

m tm g(Gm ) + 2(2t1 sm ) . (17)
Applying our assumption (16) to the inequality (17), we obtain

2
E(Gm+1 ) E(f ) E(Gm ) E(f ) C2 tq1 tm s1q 1q
m sM + 2
q+1 q q(1q)
t1 sm . (18)
We will need the following inequalities:
1. tk 1, k = 1, 2, . . .;
p 1q 1q
2. s1q
[M ]+1 ( ) sM (Lemma 2);
3. since [M ] + 1 m M , we have s[M ]+1 sm sM , and consequently,
s1q 1q
[M ]+1 sm sM .
1q
It follows from (18) that
E(Gm+1 ) E
q(1q)
E(Gm ) E C2 q+1 sM + 2q+1 (p )q(1q) sM
2(1q)
, (19)
where E := inf E(x). Let us write the chain of inequalities for all m0 from
f A1 (D)
[M ] + 1 to M , then
E(GM ) E E(Gm0 ) E (M m0 )s1q
M 1
C2 s1q 1q 1q
M (M (1 ) 1)sM 1 = sM [C2 (M (1 ) 1)1 ] , (20)
where 2 (q1)2
1 := C2 q+1 s1q
M 2
q+1 q(1q) q
sM .
Let us take any satisfying
C2 2 (q1)2
M (1)1 + 2q+1 q(1q) q sM
>
C2 q+1 s1q
M
then we obtain
E(GM ) E < 0
that can not be impossible. The smallest value of leads to a better estimate
in (14). To be sure that is smallest we can choose the parameter as follows:

C2 q+1 q(1q) q 2 (q1)
2
:= arg min + 2 sM .
01 M (1 ) 1
3.3 Dual Convergence Result for WRGA()

We need some lemmas to prove the main result.
Lemma 3. Let E be a uniformly smooth convex function defined on Banach
space X. Let (E, u) denote the modulus of smoothness of E. Then the following
inequality holds for the WRGA():
E(Gm ) E(Gm1 ) + inf (tm g(Gm1 ) + 2(E, C0 )) + , m = 1, 2, . . . ,

0
where C0 does not depend on m.

Proof. From the denition of Gm in the update step of WRGA() we have
Gm = (1 m )Gm1 + m m .
The line-search step of WRGA() implies
E(Gm ) inf E(Gm1 Gm1 + m ) + . (21)

01
It follows from Lemma 1.1 of [15] that
E(Gm1 Gm1 + m )) E(Gm1 )

E (Gm1 ), m Gm1 + 2(E, m Gm1 ). (22)
Using the gradient greedy step of WRGA() and the denition of duality gap
(7), we have
E (Gm1 ), m Gm1
tm supE (Gm1 ), s Gm1 = tm g(Gm1 ). (23)
sD
It follows from (21), (22) and (23) that
E(Gm ) E(Gm1 ) + inf (tm g(Gm1 ) + 2(E, m Gm1 )) .

0
It follows from E(Gm1 ) E(0) that Gm1 . Our assumption on bound-

ness of implies that there exists a constant C1 such that Gm1 C1 . Since
m D, we have m 1. Thus,
Gm1 m C1 + 1 =: C0 .
This completes the proof of Lemma.

Theorem 4. Let E be a uniformly smooth convex function defined on Banach
space X. Let (E, u) be the modulus of smoothness of E and suppose that
(E, u) uq , 1 < q 2. Let = {tm } m=1 , tk = , k = 1, 2, . . ., be a
1
weakness sequence. Assume that WRGA() is run for 0 < M q iterations.
Then there is an iterate 1 m M such that
g(Gm ) C(E, q, )M 1q . (24)

Proof. It follows from Theorem 2 that

1
E(Gm ) E C(E, q, )m1q , m q , (25)
where E := inf E(x). Let us suppose that
f A1 (D)
g(Gm ) > C(E, q, )M 1q (26)

for all [M ] + 1 m M , 0 < < 1 ( is xed and will be chosen later).
It follows from Lemma 3 with = m1q ,
E(Gm+1 ) E E(Gm ) E m1q tm g(Gm ) + 2(C0 m1q )q + . (27)
Using our assumption (26), the inequality (27) can be rewritten in the form
E(Gm+1 ) E
E(Gm ) E m1q tm C(E, q, )M 1q + 2(C0 m1q )q + . (28)
Since m0 m M , where m0 := [M ] + 1, the following inequalities hold:
1. m1q
0 1q M 1q ;
2. m1q
0 m1q M 1q .
Then (28) gives
E(Gm+1 ) E
E(Gm ) E C(E, q, )M 2(1q) + 2(C0 )q q(1q) M q(1q) + . (29)
If we write the chain of inequalities for all m = m0 , . . . , M , we get

E(Gm+1 ) E E(Gm0 ) E (M m0 )M 1q 2
C(E, q, )m1q
0 (M (1 ) 1)M 1q 2
C(E, q, )1q M 1q (M (1 ) 1)M 1q 2

= M 1q C(E, q, )1q (M (1 ) 1)2 ,
where

2 := M 1q C(E, q, ) 2(C0 )q q(1q) M (1q)(q1) + .
M 1q
If we take
C(E,q,)1q
M (1)1 + 2(C0 )q q(1q) M (1q)(q1) M 1q
>
M 1q C(E, q, )
then we get E(Gm ) E < 0 which is impossible. We are interested in obtaining
a better value of the constant in (24). The smallest can be attained if we
choose as follows:

C(E, q, )1q
:= arg min + 2(C0 )q q(1q) M (1q)(q1) 1q .
01 M (1 ) 1 M
4 Conclusion
Theorems 1 and 2 cited in Sect. 2 show that primal errors for the weak relaxed
greedy algorithms are small and heavily depend on geometric properties of the
objective function E. On the other hand, the paper [5] remarks that very often
both the optimal value E and the constant in the modulus of smoothness of E
are unknown, and therefore, estimates for the quality of current approximation
to optimal solution are considerably in demand. Following ideas of [5], we dened
the notion of the duality gap by the equality (7). The values of duality gap are
calculated on each iteration of both WRGA(co) and WRGA() at the gradient
greedy step, and therefore, they are inherent upper bounds for primal errors, i.e.
dierences between values of objective function at current and optimal points
on each step. We obtain dual convergence estimates for the weak relaxed greedy
algorithms in Theorems 3 and 4.
Acknowledgments. This work was supported by the Russian Fund for Basic
Research under Grant 16-01-00507. We would like to thank the reviewers profoundly
for very helpful suggestions and commentaries.
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Controlling Some Statistical Properties
of Business Rules Programs
Olivier Wang1,2(B) and Leo Liberti2

1
IBM France, 9 Rue de Verdun, 94250 Gentilly, France
2
CNRS LIX, Ecole Polytechnique, 91128 Palaiseau, France
{olivier.wang,leo.liberti}@polytechnique.edu
Abstract. Business Rules programs encode decision-making processes

using if-then constructs in a way that is easy for non-programmers to
manipulate. A common example is the process of automatic validation of
a loan request for a bank. The decision process is dened by bank man-
agers relying on the bank strategy and their own experience. Bank-side,
such processes are often required to meet goals of a statistical nature,
such as having at most some given percentage of rejected loans, or having
the distribution of requests that are accepted, rejected, and agged for
examination by a bank manager be as uniform as possible. We propose
a mathematical programming-based formulation for the cases where the
goals involve constraining or comparing values from the quantized out-
put distribution. We then examine a simulation for the specic goals of
(1) a max percentage for a given output interval and (2) an almost uni-
form distribution of the quantized output. The proposed methodology
rests on solving mathematical programs encoding a statistically super-
vised machine learning process where known labels are an encoding of
the required distribution.
Keywords: Distribution learning Mixed-integer programming Sta-

tistical goals Business Rules
1 Introduction
Business Rules (BR) are a programming for non programmers paradigm that
is often used by large corporations to store industrial process knowledge formally.
BR replaces the two most abstract concepts of programming, namely loops and
function calls, by means of an implicit outer loop and meta-variables used within
a set of easy-to-manage if-then type instructions. BR interpreters are imple-
mented by all BR management systems, e.g. [14]. BR programs are often used
by corporations to encode their policies and empirical knowledge: given some
technical input, they produce a decision, often in the form of a YES/NO output.
Corporations often require their internal processes to perform according to a
prescribed statistical behavior, which could be imposed because of strategy or
by law. This required behavior is typically independent of the BR input data.
The problem is then to parametrize the BR program so it will behave as pre-
scribed on average, while still providing meaningful YES/NO answers on given
inputs.
https://doi.org/10.1007/978-3-319-69404-7_19
264 O. Wang and L. Liberti
In [30] we studied a simplied version of the problem where the statistical

behavior was limited to a given mean. In this paper we provide a solution method-
ology for a more general (and dicult) case, where the statistical behavior is
described by a given discrete distribution. We achieve this goal by encoding a
Machine Learning (ML) procedure by means of a Mathematical Program (MP) of
the Mixed-Integer Linear Programming (MILP) type. The ML procedure relies on
non-input specic labels that encode the given knowledge about the distribution.
Controlling the statistical behavior of a complex process such as a BR program is
a very hard task, and to the best of our knowledge this work is the rst of its kind
in this respect. Methodologically speaking, we think our MILP formulation is also
innovative in that it encodes an ML training process having labels which, instead
of applying to individual inputs, apply to the entire input distribution at once.
Such an ML process bypasses the usual diculties of trying to label the training
set data, thereby being more practical for industrial applications.
The motivation for this study is a real industrial need expressed by IBM
(which co-funds this work) with respect to their BR package ODM. Our previ-
ous paper [30] laid some of the groundwork, limited to the most basic statistical
indicator (the mean of a distribution). Though that was a necessary step to the
current work, the methodology described herein is the rst to actually address
the need expressed by industry: we feel this is one of the main feature that sets
this work apart from our previous work. We still rely on MILP-based methodol-
ogy, but now the input is a whole discrete distribution, the cardinality of which
largely determines the size of the new MILP formulations presented below. Our
tests show that this has an acceptable impact on empirical solution complexity.
As an experimental illustration, we consider the two cases of the statistical
behavior being (1) a maximum percentage of a certain output value and (2) the
output values being distributed in a fashion close to the uniform distribution,
for integer outputs. We provide an optimization based approach to solving the
learning problem for each of those cases, then examine some test results.
1.1 Preliminaries
We formally represent a BR program as an ordered list of sentences of the form:
if cond(p, x) then
x act(p, x)
end if
where p is a control parameter vector (with c components) which encodes a
possible tuning of the program (e.g. thresholds which can be adjusted by the
user), x X Rd is a variable vector representing intermediate and nal stages
of computation, cond is a boolean function, and act a function with values in X.
We call rule such a sentence, condition an expression cond(p, x) and action an
instruction x act(p, x), which indicates a modication of the value of x. We
write the nal value of the variable x as xf = P (p, q), where P represents the
BR program and q is an input parameter vector representing a problem instance
and equal to the initial value of x. Although in general BR programs may have
Controlling Some Statistical Properties of Business Rules Programs 265
any type of output, we consider only integer outputs, since BR programs are
mostly used to take discrete decisions. We remark that p, x are symbolic vectors
(rather than numeric vectors) since their components are decision variables.
BR programs are executed in an external loop construct which is transparent
to the user. Without getting into the details of BR semantics, the loop executes
a single action from a BR whose condition is True at each iteration. Which BR is
executed depends on a conict resolution strategy with varying complexity. De
Sainte-Marie et al. [23] describe typical operational semantics, including conict
resolution strategy, for industrial BR management systems. In this paper, the
list of rules is ordered and the loop executes the rst BR of the list with a
condition evaluating to True at each iteration. The loop only terminates once
every condition of the BRs is False. We proved in [29] that there is a universal
BR program which can simulate any Turing Machine (TM), which makes the
BR language Turing-complete.
We consider the problem where the q Q are the past, known instances
of the BR program, and the outputs P (p, q) of those instances are divided into
N evenly sized intervals [H0 , H1 ], . . . , [HN 1 , HN ], forming a quantized output
distribution. Denoting 1 (p), . . . , N (p) the number of outputs in these categories,
we can formalize the problem as:

min p p0 1
p,x (1)
C (1 (p), . . . , N (p))
where 1 is the L1 norm and C is a constraint or set of constraints. While this

formulation uses the number of outputs rather than the probabilities themselves,
the relation between the two is simply a ratio of 1/m, where m = card(Q) is
the number of training data points.
In this paper, we suppose that P1 and P2 are BR programs with a rule set
{Rr | r } containing rules of the form:
if Lr x Gr then
x Ar x + Br
end if
with Lr , Gr , Br R and Ar {0, 1}dd . We note R = {1, . . . , } and D =
{1, . . . , d}.
We discuss the concrete example of banks using a BR program in order to
decide whether to grant a loan to a customer or not. The BR program depends
on a variable vector x and initializes its parameter vector (a component of which
is an income level threshold) to p0 . A BR program P1 is used to decide whether
a rst bank will investigate the loan request further or simply accept the auto-
mated decision taken by an expert system, and therefore has a binary output
value. This banks high-level strategy requires that no more than 50% of loans
are treated automatically, but P1 currently treats 60%. Another bank instead
uses a BR program P 2 to accept, reject, or assign a bank manager to the loan
request, and therefore has a ternary return value, represented by an integer in
{0, 1, 2}. That banks strategy requires that the proportion of each output is
{1/3, 1/3, 1/3}, but it is currently {1/4, 1/4, 1/2}. Our aim is in each case to
adjust p, e.g. modifying the income level, so that the BR program satises the
banks goal regarding automatic loan treatment. This adjustment of parameters
could be required after a change of internal or external conditions, for example.
The rst scenario can be formulated as:

min p p0 1
p,x (2)
EqQ P1 (p, q) g
where P1 has an output in {0, 1}, g [0, 1] is the desired max percentage of 1
outputs, the q Q are the past known instances of the BR program, 1 is
the L1 norm, p, q must satisfy the semantics of the BR program P (p, q) when
executed within the loop of a BR interpreter and E is the usual notation for the
expected value.
Similarly, the second scenario where P2 has an output in {1, . . . , N } and
the desired output is as close to a uniform distribution as possible can be for-
malized as:

min p p0 1
p,x (3)
s, t {1, . . . , N }, s t 1
Note that the solution to this problem is not always a truly uniform distribu-
tion, simply because there is no guarantee that m is divisible by N . However,
it will always be as close as possible to a uniform distribution, since the con-
straint imposes that all the outputs will be reached by either floor(m/N ) or
ceil(m/N ) data points. Again, we use whole numbers (of outputs in a given
interval) instead of frequencies to be able to employ integer decision variables.
Such problems could be solved heuristically by treating P1 or P2 as a black-
box, or by replacing it by means of a simplied model, such as e.g. a low-degree
polynomial. We approach this problem as in [30]: we model the algorithmic
dynamics of the BR by means of MIP constraints, in view to solving those
equations with an o-the-shelf solver. That this should be possible at all in full
generality stems from the fact that Mathematical Programming (MP) is itself
Turing-complete [16].
We make a number of simplifying assumptions in order to obtain a practi-
cally useful methodology, based on solving a Mixed-Integer Linear Programming
(MILP) reformulation of these equations using a solver such as CPLEX [13]:
1. We suppose Q is small enough that solving the MILP is (relatively) compu-
tationally cheap.
2. We assume nite BR programs with a known bound (n 1) on the number
of iterations of the loop for any input q (industrial BR programs often have
a low value of n relative to the number of rules). This in turn implies that
the values taken by x during the execution of the BR program are bounded.
We assume that M 1 is an upper bound of all absolute values of all p, q,
and x, as well as any other values appearing in the BR program. It serves as
a big M for the MP described in the rest of the paper.
3. We assume that the conditions and actions of the BR program give rise to
constraints for which an exact MILP reformulation is possible. In order to
have a linear model, each BR must thus be linear, i.e. have the form:
if L x G then
x Ax + B
end if
with L, G, B Rd and A {0, 1}dd . In general, Ah,k may have values in R
if it is not a parameter and xh has only integer values.
1.2 Related Works
We follow the formalism used in [30] pertaining to Business Rules (BR) programs
and their statistical behavior.
Business Rules (also known as Production Rules) are well studied as a knowl-
edge representation system [8,10,18], originating as a psychological model of
human behavior [20,21]. They have further been used to encode expert systems,
such as MYCIN [6,27], EMYCIN [6,25], OPS5 [5,11], or more recently ODM [14]
or OpenRules [22]. On business side of things, they have been dened broadly
and narrowly in many dierent ways [12,15,24]. We consider Business Rules as
a computational tool, which to the best of our knowledge has not been explored
in depth before.
Supervised Learning is also a well studied eld of Machine Learning, with
many dierent formulations [3,17,26,28]. A popular family of algorithms for the
classication problem uses Association Rules [1,19]. Such Rule Learning is not
to be confused with the problem treated in this article, which is more a regres-
sion problem than a classication problem. There exist many other algorithms
for Machine Learning, from simple linear regression to neural networks [2] and
support vector machines [9]. When the learner does not have as many known
output values as it has items in the training set, the problem is known as Semi-
Supervised Learning [7]. Similarly, there has been research into machine learning
when the matching of the known outputs values to the inputs is not certain [4].
A previous paper has started to explore the Learning problem when the known
information does not match to a single input [30].
2 Learning Goals with Histograms
In the rest of this paper, we concatenate indices so that (Lr )k = Lrk , (Gr )k =
Grk , (Ar )h,k = Arhk and (Br )k = Brk . We assume that rules are feasible,
i.e. r, k R D, Lk Gk . In the rest of this section, we suppose that the
dimension of p is c = 1, making p a scalar, and that p takes the place of A111 .
Similar sets of constraints exists for when the parameter p takes the place of
a scalar in Br , Lr or Gr . Additional parameters correspond to additional con-
straints that mirror the ones used for the rst parameter.
This formalization is taken from [30], in which we have also proved that the
set of constraints described in Fig. 1 models the execution of such a BR program.
The iterations of the execution loop are indexed by i I = {1, . . . , n} where n1
is the upper bound on the number of iterations, the nal value of x corresponds
to iteration n. We use an auxiliary binary variable yir with the property: yir = 1
U
i the rule Rr is executed at iteration i. The other auxiliary binary variables yir
L
and yir are used to enforce this property.
We note (C1), (C2), etc. the constraints related to the evolution of the exe-
cution and (IC1), (IC2), etc. the constraints related to the initial conditions of
the BR program:
(C1): represents the evolution of the value of the variable x
(C2): represents the property that at most one rule is executed per iteration
(C3): represents the fact that a rule whose condition is False cannot be exe-
cuted
(C4)(C6) represent the fact that only the rst rule whose condition is True
can be executed
(IC1) through (IC3) represent the initial value of a
(IC4) represents the initial value of x.
Fig. 1. Set of constraints modeling the execution of a BR program (e Rd is the

all-one vector).
2.1 A MIP for Learning Quantized Distributions
The Mixed-Integer Program from Fig. 2 models the problem from Eq. 1. We
index the instances in Q with j J = {1, . . . , m}. We also limit ourselves to
solutions which result in computations that terminate in less than n 1 rule
executions. As modifying the parameter means modifying the BR program, the
assumptions made regarding the niteness of the program might not be veried
otherwise.
We note O = {1, . . . , N }, such that t O, t = card{j J | x1n,j
[Ht1 , Ht ]}. We enforce this denition of t by using an auxiliary binary variable
stj with the property: stj = 1 i x1n,j [Bt1 , Bt ]. The other auxiliary binary
variables sU L
tj and stj are used to enforce this property.
The constraints are mostly similar to the ones in Fig. 1. We simply add the
goal of minimizing the variation of the parameter value and the constraints
C (1 (p), . . . , N (p)) from Eq. 1. The new constraints are:
(C7) represents the need for the computation to have terminated after n 1
executions
(C8)(C12) represents the denition of 1 , . . . , N
(IC4) represents (IC4) with an additional index j.
Fig. 2. Mixed-Integer Program solving Eq. 1.

That solving the MIP in Fig. 2 also solves the original Eq. 1 is a direct con-
sequence of the fact that the constraints in Fig. 1 simulate P (p, q). The proof is
simple since (C8) through (C12) trivially represent the denition of 1 , . . . , N .
A similar MIP can be obtained when p has values in dierent part of the BRs,
from which a more complex MILP is obtained for when p is non-scalar. However,
this formulation is still quite abstract, as it depends heavily on the form of C .
In fact, it can almost always be simplied given a particular constraint over the
quantized distribution, as we see in the rest of this section.
2.2 A MILP for the Max Percentage Problem
A constraint programming formulation of Eq. 2 is the Mixed-Integer Linear Pro-

gram (MILP) described in Fig. 3. In the case of the Max Percentage problem,
we can linearize the MIP in Fig. 2 as well as remove some superuous variables,
since only one of the t is relevant.
We now note e = (1, . . . , 1) Rd the vector of all ones. We use the auxiliary
2
variables w RIJR and z RIJRD such that wijr = (Ar xij + Br
xi,j )yijr (i.e. wijr = Ar xi,j + Br xi,j , the dierence between the new and the
old values of xj ) and zijrhk = arhk xi,j
k .
Any constraints numbered as before fullls the same role. The additional
constraints are:
(C11 ), (C12 ), (C13 ), (C14 ) and (C15 ) represent the linearization of (C1)
from Fig. 1
(C8) represents the goal from Eq. 2, that is a constraint over the average of
the nal values of x. It replaces C (1 , . . . , N ) and all the constraints used to
dene t from the MIP in Fig. 2.
The MILP from Fig. 3 nds a value of p that satises Eq. 2. This is again
derived from the fact that Fig. 1 simulates a BR program, and from the trivial
proof that (C11 ), (C12 ), (C13 ), (C14 ) and (C15 ) represent the linearization
of (C1).
2.3 A MILP for the Almost Uniform Distribution Problem
As before, we exhibit in Fig. 4 a MILP that solves Eq. 3. Any constraints num-
bered as before fullls the same role. The additional constraints are:
(C8) through (C10) represent the adaptation of (C8) through (C10) to the
relevant case of integer outputs
(C13) represents the equivalent to C from Eq. 3.
This MILP is obviously equivalent to solving Eq. 3, since it is for the most part
a straight linearization of the MIP in Fig. 2.
Fig. 3. MILP formulation for solving Eq. 2.
Fig. 4. MILP formulation for solving Eq. 3.

3 Implementation and Experiments

We use a Python script to randomly generate samples of 100 instances of P1
and P2 for dierent numbers of control parameters c, each instance having a
corresponding set of inputs with d = 3, n = 10 and m = 100. The number
of control parameters serves as an approximation of the complexity of the BR
program to optimize: a more complex program will have more buttons to adjust,
thus increasing the complexity, yet be more likely to have the goal be reachable
at all, i.e. have the MILP be feasible. We dene the space X as X R R Z.
The BR programs are sets of = 10 rules, where Lr , Gr , Br are vectors of
scalars in an interval range and Ar are d d matrices of binary variables. In P1 ,
we use range = [0, 1] and in P2 , we use range = [0, 3]. All input values q are
generated using a uniform distribution in range.
We use these BR programs to study the computational properties of the
MILP. The value of M used is customized according to each constraint, and is
ultimately bounded by 6 and 16 in P1 and P2 respectively (strictly greater than
ve times the range of possible values for x). We write the MILP as an AMPL
model, and solve it using the CPLEX solver on a Dell PowerEdge 860 running
CentOS Linux.
3.1 The Max Percentage Problem

We observe the proportion of solvable instances of P1 for c between 5 and 10
and c = 15 in Table 1. We use the MILP in Fig. 3 to solve Eq. 2 with the goal
set to g = 0.5.
An instance is considered solvable if CPLEX reports an integer optimal solu-
tion or a (non-)integer optimal solution. We separate the instances where the
optimal value is 0 from the others, as those indicate that the randomly generated
BR program already fulll the goal condition. We expect around fty of those
for any value of c.
In Fig. 5, we observe both the success rate and the average solving time when
considering only the non-trivial, non-timed out instances of P1 . The success rate
increases steadily, as expected. The solving time seems to indicate a non-linear
increase for c greater than 6, even with its values being somewhat unreliable due
to the small sample. Knowing that average industrial BRs are more complex
than our toy examples, regularly having thousands of rules, this approach to the
Maximum Percentage problem does not seem applicable to industrial cases.
Table 1. Experimental values for the maximum percentage problem.
Number of control parameters c 5 6 7 8 9 10 15

Trivial solvable instances (objective = 0) 52 53 49 49 58 48 46
Non-trivial solvable instances (objective = 0) 5 6 5 13 6 6 8
Infeasible instances 43 43 40 36 31 35 14
Timed out instances 0 0 7 2 5 11 32
5000
4500
4000
solving time (s)
3500
3000
2500
2000
1500
1000
500
0
5 7 9 11 13 15
number of control parameters c
Fig. 5. Average solution time over P1 for varying values of c in seconds.
3.2 The Almost Uniform Distribution Problem
We observe the proportion of solvable instances of P2 for c between 5 and 10 and

c = 15 in Table 2. We use the MILP in Fig. 4 to solve Eq. 3 with N = 2. Again,
we separate instances where the goal is already achieved before optimization,
identiable by being solved quickly with a value of p = p0 , i.e. an optimal value
of zero.
In Fig. 6, we display the success rate and average solving time over the non-
timed out, non-presolved instances for all three values of c. We observe a sharply
non-linear progression, with the average problem taking about nine minutes with
15 control parameters. Knowing that average industrial BRs are much more
complex than our toy examples, regularly having thousands of rules, we conclude
that this method can only be used infrequently, if at all.
Table 2. Experimental values for the almost uniform distribution problem.
Number of control parameters c 5 6 7 8 9 10 15

Trivial solvable instances (objective = 0) 8 2 1 1 4 7 4
Non-trivial solvable instances (objective = 0) 9 2 8 5 4 15 32
Infeasible instances 83 96 91 93 92 77 63
Timed out instances 0 0 0 1 0 1 1
600
500
solving time (s)
400
300
200
100
0
5 7 9 11 13 15
number of control parameters c
Fig. 6. Average solution time over non-trivial solvable P2 for varying values of c.
4 Conclusion, Discussion and Future Work

We have presented a learning problem of unusual type, that of supervised learn-
ing with statistical labels. We have further explored a particular subset of those
problems, those where the labels apply to a quantized output distribution. This
new approach is easily applied to practical applications in industry where control
parameters must be learned to satisfy a given goal. We have given a mathemat-
ical programming algorithm that solves such a learning problem given a linear
BR program. Depending on the specic learning problem, the mathematical
program might be easy or dicult to solve. We examined two example learn-
ing problems with practical applications for which the learning is equivalent to
solving a MILP.
We observe that, though one could detect a visual similarity in the plots
presented in Figs. 5 and 6, we believe that this similarity is only apparent. In
fact, the error bars (which measure the standard deviation of the solution time
over the instance subclass corresponding to a given size of parameters) point
out that the hard cases (with high values of solution times) are also the cases
where the error bars are longest. In other words, this similarity is simply a
result of outliers in the corresponding peaks.
The experimental results indicate the general feasibility of this type of app-
roach. It is clear that, due to the exponential nature of Branch-and-Bound (BB,
the algorithm solving the MILPs), the performance will scale up poorly with
the size of the BR program: but this can currently be said of most MILPs. This
issue, which certainly requires more work, can possibly be tackled by pursu-
ing some of the following ideas: more eective BB-based or formulation-based
heuristics (also called mat-heuristics in the literature), cut generation based on
problem structure, and decomposition. The latter, specically, looks promising
as the structure of the BR program is, up to the extent provided by automatic
translation based on parsing trees, carried over to the resulting MILP.
Other avenues of research are in extending this statistical learning approach
in other directions, e.g. learning other moments, or given quantiles in continuous
distributions. Statistical goal learning problems are an apparently unexplored
area of ML that has eminently practical applications.
Acknowledgments. The rst author (OW) is supported by an IBM France/ANRT

CIFRE Ph.D. thesis award.
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GENOPT Paper
Hybridization and Discretization Techniques
to Speed Up Genetic Algorithm and Solve
GENOPT Problems
Francesco Romito(&)
ACT Operations Research, DIAG, La Sapienza University of Rome,

Rome, Italy
francesco.romito@act-OperationsResearch.com
Abstract. One of the challenges in global optimization is to use heuristic

techniques to improve the behaviour of the algorithms on a wide spectrum of
problems. With the aim of reducing the probabilistic component and performing
a broader and orderly search in the feasible domain, this paper presents how
discretization techniques can enhance signicantly the behaviour of a genetic
algorithm (GA). Moreover, hybridizing GA with local searches has shown how
the convergence toward better values of the objective function can be improved.
The resulting algorithm performance has been evaluated during the
Generalization-based Contest in Global Optimization (GENOPT 2017), on a test
suite of 1800 multidimensional problems.
Keywords: Global optimization Discretization techniques Mixed global

local search Genetic algorithm GENOPT
1 Introduction
One of the fundamental principles in our world is the search for an optimal state.
Technological progress and the expansion of knowledge constantly bring to light the
real issues that need to be explained in quantitative terms, solving global optimization
problems as in [13]. Very often the analytical expressions of the model are missed or
are not easily represented. These problems are known in the literature as black box.
The GENOPT [4] challenge allows contestants to evaluate the algorithms on ran-
domized functions, created through suitable generators [5], provided as a binary library,
in order to be treated just as black box problems. In accord with No Free Lunch
Theorems for Optimization [6], an algorithm could reveal a positive result for a specic
benchmark function, whereas the use of randomized functions provides a major and
faithful overview of its robustness.
In literature there are now countless derivative free approaches for global opti-
mization. For instance, recently in [7] has been proposed a Deterministic Particle
Swarm Optimization, in order to better explore the search space. Another example is
given in [8] where a DIRECT-type algorithm is hybridized with a derivative-free local
minimization to globally solving optimization problems. Other well-known meta-
heuristics like Simulated Annealing [9], Tabu Search [10], Random Optimization [11],

https://doi.org/10.1007/978-3-319-69404-7_20
280 F. Romito
Ant Colony Optimization [12] are widely available in literature, and used successfully
on many real-world applications. Moreover, thorough overviews of global optimiza-
tion, with topics such as stochastic global optimization, partitioning methods, bounding
procedures, convergence studies and complexity can be found in [1316].
This paper focuses on one of the most popular classes of algorithms belonging to
evolutionary computation, namely Genetic Algorithms (GAs). Moving away from the
classical scheme [17, 18], the GA has been used as an internal procedure of a larger
scheme of global search and has been also successfully modied to reduce the prob-
abilistic component that typically characterizes it.
In particular, after the preliminary Sect. 2 about useful concepts and a look on the
complexity of the problem involved, in Sect. 3 a novel algorithmic scheme for global
optimization is presented. Ad hoc discretization techniques have been successfully
interlaced with the GA classical search operations, performing a better and wider
search on a feasible domain. Moreover, a high efcient scheme of a hybridized GA
with local searches is described. Section 4 details the tuning process and the results
obtained during the GENOPT contest, a special session of the Learning and Intelligent
Optimization Conference (LION 11: June 1921, 2017, Nizhny Novgorod, Russia).
Finally, in Sect. 5 a conclusive overview of the work is drawn, providing several lines
for further research.
2 Preliminary Concepts
The focus is on the global minimization of an N-dimensional function f x within a

hyperinterval D. The problem can be stated as follows:
f x minff x : x 2 Dg;
1
D fx 2 RN : lbi x ubi ; 1 i Ng:
No information is available on the exact analytical expression of f x, although the

belonging class is known, i.e. continuously differentiable, two time continuously dif-
ferentiable, non-differentiable or high-level features such as separable function, uni-
modal, multimodal and so on.
In reference to GENOPT rules a problem is considered solved if the solution is
identied by the algorithm within 1 million of function evaluations and so that the best
found function value is within 105 from the global minimum value:
f x f x e; x 2 D; e 105 : 2
The complexity to solve the problem (1) with an e-approximated solution is at least
exponential for any algorithm that works in the black-box model with a generic
non-convex function. On this subject, a useful result due to Vavasis [19] as a special
case of a theorem established by Nemirovsky and Yudin [20] is explanatory:
Theorem 1. Let F k; p be the class of k-times differentiable functions on D whose kth
derivative is bounded by p as follows: at any point x 2 D and for any unit vector u,
Hybridization and Discretization Techniques 281
k
d
f x tu p: 3
dtk
Let A be any minimization algorithm that works in the black-box model (evaluating
f and its derivatives). Assume that for any function 2 F k; p, A is guaranteed to output
a point that satises inequality (2). Then, there exists a function f 2 F k; p such that
algorithm A will run on f for at least a number of steps given
pn=k
c ; 4
e
with c being a suitable positive constant.
As mentioned before, what emerges is that, even assuming bounds on the
derivatives, the complexity of solving a global minimization problem increases
exponentially with the problems dimension.
Usually the problem (1) cannot be solved by an exhaustive search algorithm in an
efcient time. Next section will introduce an approach based on the idea of space
search reduction to lead the search towards the most promising area.
3 The GABRLS Algorithm
Subsection 3.1 is aimed to the description of the modied Genetic Algorithm (GA),
while in Subsect. 3.2 a novel Bounding Restart (BR) technique is described. Addi-
tionally, addressing the need to improve the convergence speed, an overall scheme with
derivative free Local Searches (LS) is presented in Subsect. 3.3.
3.1 The Modied GA

Starting from a classical pattern of GA (see Algorithm 1), to take advantage of the
effective geometry and to prevent premature convergence, two major changes were
introduced:
Algorithm 1. Classical GA scheme
1. Initial Population (random points uniformly distributed in D)

2. for k = 1 generations (max iterations)
3. Evaluate Population
4. Selection Criterion (Tournament, Elitism, Roulette wheel, etc.)
5. Genetic Operators (Crossover Mutation)
6. New Population
7. end for
282 F. Romito
Discrete initial population

Diagonal Initial Population (DiagPI).
Axial Initial Population (AxialPI).
Premature convergence preserving procedure
Diversify.
The rst change concerns the implementations of two discretization and positioning
techniques to place the initial points (initial population) in the feasible domain,
allowing the sampling of the objective function in such a way to get more useful
information than a classical random sampling.
DiagPI routine allows the discrete positioning of points along a main diagonal of
the feasible hyperrectangle (Fig. 1 provides an example in a 3D box with a vertex in
the origin of the coordinate axes). Denoting with Spacej the amplitude of the interval
along the jth dimension, with Increasej the step along the jth dimension between two
consecutive points and with Pop the number of initial points where the objective
function f is evaluated, the result is the Algorithm 2.
Fig. 1. Graphic view of the points (blue squared) generated with DiagPI in a 3D box. (Color
gure online)
Algorithm 2. DiagPI
1. for j = 1 N
2.
3.
4. end for
5. for i = 1 Pop
6. for j = 1 N
7.
8. end for
9. end for
Fig. 2. Graphic view of the points (red squared) generated with AxialPI in a 3D box. (Color
gure online)
284 F. Romito
DiagPI and AxialPI routines allow to create, through the crossover operator during
the iterations of GA (main generations loop), a multi-dimensional mesh with the points
generated as nodes (Fig. 3 provides an example in a 3D box with a vertex in the origin
of the coordinate axes).
AxialPI routine distributes the points of population along the coordinate axes of the
feasible hyperinterval (Fig. 2), the result is the Algorithm 3.
Fig. 3. Mesh of all possible points generated by the crossover (grey) through the recombination
of the initial ones (AxialPI in red, DiagPI in blue). (Color gure online)
This mesh will become more dense according to the most promising areas of the
feasible domain D. In particular, after every selection phase in a GA iteration, a
single-point crossover operator has been adopted, without recombination
of blocks to
generate and place new points PSon i j : i 1 ! Pop; j 1 ! N in the mesh.
Equations (5) and (6) and Fig. 4 show how it works.

PSon
1 PFather
1 1; . . .; PFather
1 k 1; PFather
2 k; . . .; PFather
2 N ; 5

PSon
2 PFather
2 1; . . .; PFather
2 k 1; PFather
1 k; . . .; PFather
1 N : 6
Fig. 4. Single-point crossover operator without recombination of blocks (exact exchange).

With respect to a random sampling, the discrete positioning techniques described

above, together with the single-point crossover operator without recombination of
blocks, allow the exploration of the search space D in a more orderly manner, and also
controlled, the feasible region. In fact, all possible positions in which the crossover will
be able to move the points are known a priori and are all nodes of a multi-dimensional
mesh.
The second change of GA classical scheme concerns the implementation of a
simple technique that allows one to avoid a premature convergence.
Inside the main loop of Algorithm 1 a check on homogeneity of current population
has been inserted (Algorithm 4). With the aim of pursuing an aggressive search, a
routine (hereinafter called Diversify) makes a more extensive search by replacing all
the duplicate points of the current population with new points PNew j; j 1 ! N .
Uniformly distributed random numbers in the range lb; ub have been used to mutate
all components j 1 ! N of the duplicate points, so that new random points are
placed in the feasible domain. As drawback, the worst case computational cost of this
routine is generally high, i.e. a simple implementation can take Pop 12 2 N
steps, so it is reasonable to carry out a check on homogeneity of population after at least
half of the iterations of the GA main loop.
3.2 Bounding Restart (BR) Technique

Generally, increasing the dimension of problems and the search space, GA takes more
function evaluations to locate a good solution. To avoid this slowness an iterative space
reduction technique has been implemented.
BR technique is a two-step scheme in which GA can be successfully integrated.
The rst step is the bounding one. If it is assumed that the genetic algorithm has the
ability to quickly identify a promising area, however large, its reasonable to focus the
search in the given area temporarily. On that basis, by means of hyperintervals that are
dynamically sized, according to the need to lead the search towards the most promising
286 F. Romito
area, the bounding step at the generic iteration k is carried out through the following
equations:
ub lb ub lb
LBk ; CF 2 R : CF const [ 1; expCF 2 N: 7
2 2 CF expCF
ub lb ub lb
UBk ; CF 2 R : CF const [ 1; expCF 2 N: 8
2 2 CF expCF
CF is a convergence factor that has a high impact on reducing the bounds. The
reduction is managed by increasing expCF, the exponent of CF, of a unit per iteration.
After updating lower and upper bounds, the reduced set is centred in the best point xk
currently known. Let Ctraslk be the difference between xk and the centre of the kth
reduced hyperinterval, then, the set can be put centrally as follows:
UBk LBk
Ctraslk xk ; 9
2

LBk 1 max lb; LBk Ctraslk ; 10
UBk 1 minfub; UBk Ctraslk g: 11
At the kth BR reduction cycle, assuming expCF k the feasible space reduced is:
ub lb
Spacek [ d; d machine precision: 12
CF k
In Fig. 5 the result of a bounding and positioning operation in a 3D box is drawn.
Fig. 5. Overview of a bounding step of BR. In red Pi , the best solution currently known. (Color
gure online)
The second step of BR concerns the restart of GA inside the reduced space.
Denoting with Dk the kth reduced hyperinterval, the overall algorithmic scheme is the
following:
3.3 Hybridizing GABR with Local Searches

The global search represented by Algorithm 5 can be expensive in terms of function
evaluations if the goal is to identify an optimal solution with a high precision. The
reason is that the higher is the precision required, the more BR iterations must be
performed. The rening of the solution can be delegated to a local search algorithm.
In order to make the whole algorithm more efcient, derivative free local searches
(DFLS) have been introduced. In literature there are many ideas on the hybridization of
the global search. Examples of automatic balancing techniques can be found in [21]
where the estimates of local Lipschitz constants allow to accelerate signicantly the
global search. A suitable strategy has been implemented in the GABR algorithm,
similar to that one described in [22].
Since the number of LSs performed affects the efciency, it is necessary to locate
when a LS should be started. A reasonable choice is to perform LS at every BR
iteration, only after the end of GA main loop and only if an improved solution is found
with respect to previous BR iteration.
Let us consider as stopping criterion the GENOPT condition of 106 max function
evaluations. The nal algorithmic scheme is the following:
288 F. Romito
4 Tuning and Results of GABRLS on GENOPT Challenge
GENOPT organizers provide 1800 problems, distinguishable in 18 classes by their

dimension and other high-level characteristics.
This section will report the most important settings of GABRLS algorithm that
have impacted on the progressive improvement in results during the challenge.
Let us consider that DiagPI, AxialPI and Diversify routines are always active
hereinafter in the GABRLS algorithm.
4.1 High Level Setting

The rst strategy was the selection of a suitable local search to evaluate the
improvement on convergence toward a better approximation of the global optimum
with respect to the GABR scheme without LSs. In the preliminary phase of challenge, a
non-monotone local search has been employed. In particular, a version of the globally
convergent coordinate search algorithm called SDBOX [23] has been used, freely
available in different programming languages in the software library for derivative free
optimization at http://www.diag.uniroma1.it/*lucidi/DFL.
Table 1 details the results of the comparison, reports the number of global optima
found over the hundred problems of each class of function. Clearly, it came out as
expected that using local searches the behaviour of the algorithm is improved.
Subsequently another local search, integrated into Matlab toolbox (FMINCON
[24]), has been tested. In the latest conguration of the GABRLS only FMINCON was
Table 1. Number of the problems solved by the algorithms with and without LS
Id Type Type-details Dim GABRLS GABR
Tasks solved Tasks solved
0 GKLS Non-differentiable 10 92 92
1 30 59 40
2 Differentiable 10 73 74
3 30 45 27
4 Twice differentiable 10 67 72
5 30 43 34
6 High condition Rosenbrock 10 100 3
7 30 71 0
8 Rastrigin 10 100 99
9 30 100 0
10 Zakharov 10 100 100
11 30 2 0
12 Composite 10 100 7
13 30 100 0
14 10 100 3
15 30 100 0
16 10 100 69
17 30 100 0
Total 1452 620
integrated as LS with default parameters because of the slightly better result on several
classes of function. In particular, FMINCON and NM-SDBOX showed equal perfor-
mance to identify a local optimum on GKLS (classes 0, , 5), whereas FMINCON
outperformed NM-SDBOX on all remaining classes.
The second high level strategy was to tune the amount of population and genera-
tions of GA. These parameters give an impact on the search performance and are
crucial to balance efciency (CPU time and convergence speed) and effectiveness (fast
identication of the global optimum neighbourhood). Smaller values lead quickly to
less quality solutions, while larger values allow to identify a more promising areas but
slowing down the search.
The main aim was to nd out the smallest value that allows to improve efciency
and solve the maximum number of tasks. The setting of these parameters on GKLS
classes appeared more sensitive than other classes, so a mixed strategy has been
implemented. In particular, two settings have been adopted as starting point of tuning,
to solve GKLS functions and all other classes of functions, respectively.
Table 2 reports the selection phase of the two starting settings of population and
generations indexed by scenarios.
Table 2. Selection of the starting setting of GA. The best ones are highlighted.
The nal amount of Population and Generations are rened for each class of
GENOPT problems, so small changes have been made from the two guideline of
scenarios selected.
The other high level parameters of GABRLS algorithm are self tuned or constants.
After every selection phase of GA, carried out through the efcient well known
Tournament Selection [25], a Random Mutation [26] with xed rate has been inte-
grated, as usual, inside the crossover operator to insure that the probability of reaching
any point in the search space is never equal to zero. Table 3 reports the starting value
and the updating rule of the most important parameters.
Table 3. Setting of other high level GA and BR parameters.

Parameter Starting Updating formula Frequency
CF 1.2 CF CF 0:01 BR iterations
expCF 1 expCF iBR; iBR ith BR cycle BR iterations
Crossover Rate 0.8 Constant GA iterations
Mutation Rate 0.01 Constant GA iterations
Tournament 3 Constant GA iterations
290 F. Romito
4.2 Results and Prizes

In the nal phase of the GENOPT competition, GABRLS algorithm was able to reach
the 1st prize in both partial categories. Figure 6 shows the score on the speed of
convergence (High Jump), the task solved (Target Shooting) and the overall ranking
(Biathlon Score).
Fig. 6. Final leaderboard.
The total number of solved problem was 1605 over 1800, almost 90%. Table 4
presents the number of solved problem for each class of function.
Table 4. Number of the solved tasks for each class.

Id Type Type-details Dim GABRLS
Tasks solved
0 GKLS Non-differentiable 10 86
1 30 74
2 Differentiable 10 77
3 30 56
4 Twice differentiable 10 66
5 30 47
6 High condition Rosenbrock 10 100
7 30 100
8 Rastrigin 10 99
9 30 100
10 Zakharov 10 100
11 30 100
12 Composite 10 100
13 30 100
14 10 100
15 30 100
16 10 100
17 30 100
Total 1605
5 Conclusion
In this work, a novel algorithmic scheme for global optimization is presented. Several
discretization techniques to place initial points and to bound the search space are
described. The numerical results show that the overall scheme with local searches step
up the effectiveness on locating optimal solutions within a high precision range.
Further research will involve on improving the exploratory geometry through the
use of different bounding procedure. Moreover, algorithmic schemes for
multi-objective optimization and constrained optimization will be investigated.
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generalization-based challenge in global optimization. In: Sergeyev, Y.D., Kvasov, D.E.,
DellAccio, F., Mukhametzhanov, M.S. (eds.) AIP Conference Proceedings, vol. 1776, no.
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41000-5_32
8. Di Pillo, G., Liuzzi, G., Lucidi, S., Piccialli, V., Rinaldi, F.: A DIRECT-type approach for
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Short Papers
Identication of Discontinuous Thermal
Conductivity Coefcient Using Fast
Automatic Differentiation
Alla F. Albu1,2, Yury G. Evtushenko1,2, and Vladimir I. Zubov1,2(&)

1
Dorodnicyn Computing Centre, Federal Research Center
Computer Science, and Control of Russian Academy of Sciences,
Moscow, Russia
alla.albu@yandex.ru, evt@ccas.ru,
vladimir.zubov@mail.ru
2
Moscow Institute of Physics and Technology,
Dolgoprudny, Moscow Region, Russia
Abstract. The problem of determining the thermal conductivity coefcient that

depends on the temperature is studied. The consideration is based on the
Dirichlet boundary value problem for the one-dimensional unsteady-state heat
equation. The mean-root-square deviation of the temperature distribution eld
and the heat flux from the empirical data on the left boundary of the domain is
used as the objective functional. An algorithm for the numerical solution of the
problem based on the modern approach of Fast Automatic Differentiation is
proposed. The case of discontinuous thermal conductivity coefcient is con-
sidered. Examples of solving the problem are discussed.
Keywords: Inverse coefcient problems Variation problem Fast Automatic

Differentiation Discontinuous thermal conductivity coefcient Heat equation
1 Introduction
The classical heat equation is often used in the description and mathematical modeling
of many thermal processes. The density of the substance, its specic thermal capacity,
and the thermal conductivity coefcient appearing in this equation are assumed to be
known functions of the coordinates and temperature. Additional boundary value con-
ditions makes it possible to determine the dynamics of the temperature eld in the
substance under examination.
However, the substance properties are not always known. It often happens that the
thermal conductivity coefcient depends only on the temperature, and this dependence
is not known. In this case, the problem of determining the dependence of the thermal
conductivity coefcient on the temperature based on experimental measurements of the
temperature eld arises. This problem also arises when a complex thermal process
should be described by a simplied mathematical model. For example, in studying and
modeling of heat propagation in complex porous composite materials, where the
radiation heat transfer plays a considerable role, both the convective and radiative heat

https://doi.org/10.1007/978-3-319-69404-7_21
296 A.F. Albu et al.
transfer must be taken into account. The thermal conductivity coefcients in this case
typically depend on the temperature. To estimate these coefcients, various models of
the medium are used. As a result, one has to deal with a complex nonlinear model that
describes the heat propagation in the composite material (see [1]). However, another
approach is possible: a simplied model is constructed in which the radiative heat
transfer is not taken into account, but its effect is modeled by an effective thermal
conductivity coefcient that is determined based on empirical data.
Determining the thermal conductivity of a substance is an important problem, and it
has been studied for a long time. This is conrmed by a large number of publications
(e.g., see [25]).
In [6] the inverse coefcient problems are considered in new formulation. They are
studied theoretically, and new numerical methods for their solution are developed. In
that work the case of continuous thermal conductivity coefcient is considered.
In this paper we consider the problem studied in [6] for the case of a discontinuous
thermal conductivity coefcient. The consideration is based on the Dirichlet problem
for the one-dimensional unsteady-state heat equation. The inverse coefcient problem
is reduced to a variation problem. A linear combination of the mean-root-square
deviations of the temperature distribution eld and the heat flux from the empirical data
on the left boundary of the domain is used as the objective functional. An algorithm for
the numerical solution of the inverse coefcient problem is proposed. It is based on the
modern approach of Fast Automatic Differentiation, which made it possible to solve a
number of difcult optimal control problems for dynamic systems.
2 Formulation of the Problem
A layer of material of width L is considered. The temperature of this layer at the initial
time is given. It is also known how the temperature on the boundary of this layer
changes with time. The distribution of the temperature eld at each moment of time is
described by the following initial boundary value (mixed) problem:

@Tx; t @ @Tx; t
qC KT 0; x; t 2 Q 1
@t @x @x
Tx; 0 w0 x; 0 x L; 2
T0; t w1 t; TL; t w2 t; 0 t H: 3
Here x is the Cartesian coordinate of the point in the layer; t is time,

Q f0 \ x \ L 0 \ t Hg; Tx; t is the temperature of the material at the
point with the coordinate x and time t; q and C are the density and the heat capacity of
the material, respectively; KT is the coefcient of the convective thermal conduc-
tivity; w0 x is the given temperature of the layer at the initial time; w1 t is the given
temperature on the left boundary of the layer; and w2 t is the given temperature on the
right boundary of the layer.
Identication of Discontinuous Thermal Conductivity Coefcient 297
The density of the material q and its heat capacity C are known functions of
coordinate and/or temperature.
If the dependence of the coefcient of the convective thermal conductivity KT on
the temperature T is known, then we can solve the mixed problem (1)(3) to nd the
Problem (1)(3) will be further called the
distribution of the temperature Tx; t in Q.
direct problem.
If the dependence of the coefcient of the convective thermal conductivity of the
material on the temperature is not known, it is of interest to determine this dependence.
A possible statement of this problem (it is classied as an identication problem of the
model parameters) is as follows: nd the dependence KT on T under which the
temperature eld Tx; t obtained by solving problem (1)(3) is close to the eld
Yx; t, which itself is obtained empirically. The quantity
ZH ZL
UKT Tx; t Yx; t 2 lx; t dxdt
0 0
4
ZH 2
@T
b K T0; t 0; t Pt dt;
@x
0
can be used as the measure of difference between these functions. Here b 0 is a given
number, lx; t 0 is a given weighting function, and Pt is the known heat flux on
the left boundary of the domain. Thus, the optimal control problem is to nd the
optimal control KT and the corresponding optimal solution Tx; t of problem (1)(3)
that minimizes functional (4).
3 Numerical Solution of the Problem
The optimal control problems similar to this one are typically solved numerically using
a descent method, which requires the gradient of functional (4) to be known. The
unknown function KT was approximated by a continuous piecewise linear function.
If the input data of the problem is such that the desired coefcient of thermal
conductivity represents a fairly smooth function, then the problem of identication
could be solved by a method proposed in [6].
This work presents the algorithm for solving the problem of identifying a dis-
continuous thermal conductivity coefcient and some numerical results. The proposed
algorithm, as well as in [6], is based on the numerical solution of the problem of
minimizing the cost functional (4). The gradient descent method was used. It is well
known that it is very important for the gradient methods to determine accurate values of
the gradients. For this reason, we used the efcient approach of Fast Automatic Dif-
ferentiation that enables us to determine with machine precision the gradient of cost
function, subject to equality constraints (see [7]).
To numerically solve the mixed problem (1)(3) the domain Q f0\x\L
0\t Hg is decomposed by the grid lines f~xi gIi0 and f~t j gj0 into rectangles.
J
At each node ~xi ; ~t j of Q characterized by the pair of indices i; j, all the functions are
determined by their values at the point ~xi ; ~t j (e.g., T ~xi ; ~t j Tij ). In each rectangle,
the thermal balance must be preserved.
The temperature interval a; b (the interval of interest) is partitioned by the points,
T0 a, T1 ; T2 ; . . .; TN b into N 2m 1 parts (they can be of equal or of different
lengths). Each point Tn n 0; . . .; N is assigned a number kn KTn . The function
KT T, which needs to be found, is approximated by a continuous piecewise linear
N
functions with the nodes at the points T~n ; kn so that
n0
kn kn1
KT kn1 T Tn1 for Tn1 T Tn ; n 1; . . .; N:
Tn Tn1
The objective functional (4) was approximated by a function Fk0 ; k1 ; . . .; kN of

the nite number of variables as
X I1
J X
UKT F Tij Yij 2 lij hi s j
j1 i1
XJ
r

b KT0j KT1j T1j T0j
j1
2h0
2 !
1 r qC h
0 0

KT0j1 KT1j1 T1j1 T0j1 0 j T0j T0j1 P j s j :
2h0 2s
To illustrate the efciency of the proposed algorithm the variation problem (1)(4)
was considered with the following parameters:
L 1; H 1; qx Cx 1;
w0 x 2x; w1 0; w2 2;
lx; t 1; b 0; a 0; b 2:
The empirical temperatureeld was determined as a solution of the direct

1; T\1;
problem (1)(3) with KT .
3; T 1:
On the rst m subintervals KT 1, on the m 2; m 3; . . .; N subintervals
KT 3, and on the m 1 subinterval KT is a linear function varying in from 1 to 3.
For the solution of the direct and inverse problems, we used the uniform grid with
the parameters I 300 (the number of intervals along the axis x) and J 6000 (the
number of intervals along the axis t), which ensures the sufcient accuracy of com-
putation of the temperature eld.
Figure 1 shows the results (the initial approximation and optimal solution) of
calculations made for N 19. As an initial approximation to the thermal conductivity
coefcient the function KT T was selected.
The value of the cost function (4) for initial approximation was equal to
Uini 4:241570 100 . The value of the cost function (4) for optimal control was equal
Identication of Discontinuous Thermal Conductivity Coefcient 299
Fig. 1. Initial and optimal control N 19
to Uopt 1:013323 1021 . In this example, we managed to identify the thermal

conductivity coefcient with precision 109 in norm C.
Figure 2 presents the same results of calculations as above but made for N 79 (the
initial approximation and optimal solution). The value of the cost function (4) for initial
approximation was equal to Uini 9:532799 103 . The value of the cost function (4)
for optimal control was equal to Uopt 2:288689 1022 . In this case, we managed to
identify the thermal conductivity coefcient with precision 1010 in norm C.
Fig. 2. Initial and optimal control (N 79)
It should be noted that in this example, the thermal conductivity coefcient being
identied was a continuous function, although contained narrow domain of smooth-
ing jump (its width is 2=N).
The numerous numerical results showed that the proposed algorithm is efcient and
stable and allows us to restore thermal conductivity with high accuracy.
Acknowledgments. This work was supported by the Russian Foundation for Basic Research
(project no. 17-07-00493a).
References
1. Alifanov, O.M., Cherepanov, V.V.: Mathematical simulation of high-porosity brous
materials and determination of their physical properties. High Temp. 47, 438447 (2009)
2. Kozdoba, L.A., Krukovskii, P.G.: Methods for Solving Inverse Thermal Transfer Problems.
Naukova Dumka, Kiev (1982). [in Russian]
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[in Russian]
4. Marchuk, G.I.: Adjoint Equation and the Analysis of Complex Systems. Nauka, Moscow
5. Samarskii, A.A., Vabishchevich, P.N.: Computational Heat Transfer. Editorial URSS,
Moscow (2003). [in Russian]
6. Zubov, V.I.: Application of fast automatic differentiation for solving the inverse coefcient
problem for the heat equation. Comput. Math. Math. Phys. 56(10), 17431757 (2016)
7. Evtushenko, Y.G.: Computation of exact gradients in distributed dynamic systems. Optim.
Methods Softw. 9, 4575 (1998)
Comparing Two Approaches for Solving
Constrained Global Optimization Problems
Konstantin Barkalov(B) and Ilya Lebedev

{konstantin.barkalov,ilya.lebedev}@itmm.unn.ru
Abstract. In the present study, a method for solving the multiextremal

problems with non-convex constrains without the use of the penalty or
barrier functions is considered. This index method is featured by a sep-
arate accounting for each constraint. The check of the constraint fulll-
ment sequentially performed in every iteration point is terminated upon
the rst constraint violation occurs. The results of numerical comparing
of the index method with the penalty function one are presented. The
comparing has been carried out by means of the numerical solving of sev-
eral hundred multidimensional multiextremal problems with non-convex
constrains generated randomly.

convex constraints
1 Introduction
In the present paper, the methods for solving the global optimization problems
with non-convex constraints
(y ) = min {(y) : y D, gi (y) 0, 1 i m}, (1)

D = y RN : aj yj bj , 1 j N . (2)
are considered. The objective function as well as the constraint ones are supposed
to satisfy Lipschitz condition with Lipschitz constants unknown a priori. The
analytical formulae of the problem functions may be unknown, i.e. these ones
may be dened by an algorithm for computing the function values in the search
domain (so called black-box-functions). Moreover, it is suggested that even a
single computing of a problem function value may be a time-consuming operation
since in the applied problems it is related to the necessity of numerical modeling
(see, for example, [14]).
The method of penalty functions is one of the most popular numerical meth-
ods for solving the problems of this kind. The idea of the method is simple and
very universal; therefore, the method has found wide application to solving var-
ious practical problems. A detailed description of the method can be found, for
https://doi.org/10.1007/978-3-319-69404-7_22
302 K. Barkalov and I. Lebedev
example, in [5]. Nevertheless, the method of penalty functions has a number of

disadvantages.
First, when using this method, a number of unconstrained problems with
dierent penalty coecients should be solved. The issue of choosing the penalty
coecients should be addressed for each problem separately.
Second, the use of large values of the penalty coecients results in the min-
imization of the functions with a ravine-type minimum that essentially compli-
cates the computations.
Third, this method is not applicable in the problems with the partially com-
putable functions, for example, in the case when the objective function is unde-
ned if the constraints are violated.
In the present work, the use of the penalty function method is compared to
the application of the index method of accounting for the constraints developed
in Nizhny Novgorod State University. The method features are (i) accounting
for the information on each constraint separately and (ii) solving the problems,
in which the function values may be undened out of the feasible domain.
2 Index Method
A novel approach to the minimization of the multiextremal functions with non-
convex constraints (called the index method of accounting for the constraints)
has been developed in [69]. The approach is based on a separate accounting
for each constraint of the problem and is not related to the use of the penalty
functions. At that, employing the continuous single-valued Peano curve y(x)
(evolvent) mapping the unit interval [0, 1] on the x-axis onto the N -dimensional
domain (2) it is possible to nd the minimum in (1) by solving a one-dimensional
problem
(y(x )) = min {(y(x)) : x [0, 1], gi (y(x)) 0, 1 i m} .
The considered dimensionality reduction scheme juxtaposes to a multidimen-

sional problem with Lipschitzian functions a one-dimensional problem, where
the corresponding functions satisfy uniform Holder condition (see [6]).
According to the rules of index method, every iteration called trial at corre-
sponding point of the search domain includes a sequential checking of fulllment
of the problem constraints at this point. The rst occurrence of violation of any
constraint terminates the trial and initiates the transition to the next iteration
point, the values of the rest problem functions are not computed in this point.
This allows: (i) accounting for the information on each constraint separately and
(ii) solving the problems, in which the function values may be undened out of
the feasible domain.
The index algorithm can be applied to solving the unconstrained problems as
well, i.e. in the case when m = 0 in the problem statement (1). In this case, the
algorithm works analogously to its prototype global search algorithm (GSA)
developed for solving the unconstrained optimization problems. Various modi-
cations of the index algorithm and the corresponding theory of convergence are
Comparing Two Approaches for Solving Constrained Global Optimization 303
presented in [6]. The algorithm is very exible and allows an ecient paralleliza-
tion as for shared memory, as for distributed memory, as for accelerators [1014].
3 Results of Experiments
Let us compare the solving of the constrained global optimization problems using
penalty functions method (PM) and index method (IM). The penalty function
was taken in the form
m
2
G(y) = max {0, gj (y)} ,
j=1
where is the penalty coecient. Thus, a constrained problem is reduced to an

unconstrained problem
F (y) = (y) + G(y)
which was solved using global search algorithm.
The experiments was carried out with the use of GKLS generator. This
generator for the functions of arbitrary dimensionality with known properties
(the number of local minima, the size of their domains of attraction, the global
minimizer, etc.) has been proposed in [16]. Eight GKLS classes of dierentiable
test functions of the dimensions N = 2, 3, 4, and 5, have been used. For each
dimension, both Hard and Simple classes have been considered. The diculty
of a class was increased either by decreasing the radius of the attraction region
of the global minimizer, or by decreasing the distance from the global minimizer
y to the domain boundaries. Application of this generator for studying some
unconstrained optimization algorithms has been described in [15,1719].
According to the scheme described in [20], eight series of 100 problems each
based on the functions of Simple and Hard classes with the dimensions N =
2, 3, 4, 5 have been generated. There were two constraints and objective functions
in each problem. The volume fraction of the feasible domain was varied from 0.2
to 0.8. The penalty parameter was selected to be equal to 100.
The global minimum was considered to be found if the algorithm
generates a
trial point y k in the -vicinity of the global minimizer, i.e. y k y . The
size of the vicinity was selected as = 0.01 b a. When using IM for Simple
class, parameter r = 4.5 was selected, for Hard class r = 5.6 was taken. The
maximum allowed number of iterations was Kmax = 106 .
The average number of iterations kav performed by PM for solving a series
of problems from both classes is shown in Table 1. The same values for the
index method are presented in Table 2. The number of unsolved problems is
specied in brackets. It reects a situation when not all problems of the class
have been solved by the method. This means that the algorithm was stopped as
the maximum allowed number of iterations Kmax was achieved. In this case, the
value Kmax = 106 was used for calculating the average value of the number of
iterations kav that corresponds to the lower estimate of the average value.
Table 1. The values of kav when solving the GKLS problems by PM
N =2 N =3 N =4 N =5
Simple Hard Simple Hard Simple Hard Simple Hard
0.2 1639 1507 56149 60651 160198(5) 119764(2) 305957(1) 488587(14)
0.4 1534 1967 67145 74860 158547 127832 396480(4) 434328(2)
0.6 1201 1514 92854 101240 138550 143818 373617(8) 561447(9)
0.8 1277 1287 108121 148479 130372 145592 488791(12) 646538(24)
Table 2. The values of kav when solving the GKLS problems by IM
N =2 N =3 N =4 N =5
Simple Hard Simple Hard Simple Hard Simple Hard
0.2 447 911 14719 20120 59680 66551(1) 391325(2) 188797(12)
0.4 465 1800 11951 17427 71248 86899 339327(1) 151998(3)
0.6 403 1988 7366 12853 58451 92007 316648 179013(4)
0.8 371 4292 4646 8702 33621 54405 309844 124952
Table 3. Solving of GKLS problems by the index method, N = 4
Simple Hard
k1 k3 k3 k1 k2 k3
0.2 59680 20445 4401 66551 24210 6316
0.4 71248 28527 6784 86899 39682 12615
0.6 58451 31508 9505 92007 52560 19853
0.8 33621 21411 10446 54405 36838 22202
The average values presented demonstrate that the solving of the specied
problems using the index method requires less number of iterations than with
the use of the penalty function one. At the same time, separate accounting for
the constraints in the index method provides less number of computations of
the values of the problem functions as well. The numbers of computations of
the values of the constraint functions g1 (y), g2 (y) and those for the objective
function (y) (k1 , k2 , and k3 , respectively) are presented in Table 3 for four-
dimensional problems.
4 Conclusion
Concluding, let us note that the index method for solving constrained global
optimization problems considered in the present work:
is based on the global search algorithm, which is not inferior in the speed of
work than other well-known algorithms;
Comparing Two Approaches for Solving Constrained Global Optimization 305
allows solving the initial problem directly, without the use of the penalty
functions (thus, the issues of selection the penalty coecient and of solv-
ing a series of unconstrained problems with dierent penalty coecients are
eliminated);
allows solving the problems, which the values of the problem function are not
dened everywhere (for example, the objective function values are undened
out of the problem feasible domain);
speeds up the process of solving the constrained optimization problems (due
to an essential reduction of the total number of computations of the problem
function values).
The last statement has been supported by the numerical solving several hundred
test problems.

project No 16-11-10150.
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10556, pp. 314319. Springer, Cham (2017). doi:10.1007/978-3-319-69404-7 24
Towards a Universal Modeller
of Chaotic Systems
Erik Berglund(B)
Jeppesen, Gothenburg, Sweden

erik.berglund@jeppesen.com
Abstract. This paper proposes a Machine Learning (ML) algorithm

and examines its dynamic properties when trained on chaotic time series.
It will be demonstrated that the output of the ML system is itself more
chaotic if it is trained on a chaotic input than if it is trained on non-
chaotic input. The proposed ML system builds on to the Parameter-Less
Self-Organising Map 2 (PLSOM2) and introduces new developments.
Keywords: Machine learning Self-organization Chaotic Recur-

rence Recursion Universal model
1 Introduction
The class of dynamical systems that are non-linear and highly sensitive to initial
conditions - commonly called chaotic - appear in many places in nature, and it
has even been suggested that chaos plays a part in biological intelligence [6] and
perception [7]. Since deterministic systems can be chaotic (indeed, this is where
the phenomenon was rst discovered), chaos is of interest to computer scientists.
Chaotic systems have been subject of study in the Machine Learning com-
munity. Much, if not most, of this eort has been aimed at predicting the time
evolution of chaotic systems based on starting conditions and a manually con-
structed model of the chaotic system. This prediction is, by the very nature of
chaotic systems, extremely dicult and the predictive value of the model quickly
diminishes as the prediction time period increases.
Instead this paper presents a machine learning system that, by being trained
on the output of a dynamical system, can replicate some of the fundamental
properties of the dynamical system without necessarily attempting to predict
the evolution of the dynamical system itself, thus creating a tool that can model
chaotic behaviour without knowing the underlying rules of the chaotic system.
The algorithm is trained on four dierent deterministic dynamical processes,
of which one is chaotic and the others are not, and the chaotic properties of the
system are measured.
The rest of this document is laid out as follows: Sect. 2 discusses previous
works and gives background information, Sect. 3 gives details for the machine
learning system, Sect. 4 describes the experimental setup and Sect. 5 lists the
results. Section 6 contains concluding remarks and discussion.
https://doi.org/10.1007/978-3-319-69404-7_23
308 E. Berglund
2 Previous Work
Chaotic time-series and chaos in neural networks has been studied by several
researchers. One prominent example is CSA [4], which has been shown to be
able to solve combinatorial tasks like the Travelling Salesperson Problem (TSP)
eciently. Unlike the present work the chaos is inherent in the network model,
not learned. CSA has inspired other approaches, for example [13], which also
contains a good review of similar methods.
Time-series processing with SOM-type algorithms have been studied before,
with the aim of predicting time series [3,8].
Another avenue of research is the role of chaotic neural networks in memory
formation and retrieval [5].
The nomenclature of SOM algorithms with recurrent connections is confus-
ing. In other ML algorithms recurrence usually means that the output of the
algorithm is fed back as part of the input. The Recurrent SOM [11], on the
other hand, is in essence a SOM with leaky integrators on the inputs. When
recurrent connections in SOM algorithms were rst investigated [12], the term
recursive was used instead.
Self-Organising Maps are a form of unsupervised learning, where a set of
nodes indexed by i [1, n] are each associated with a weight vector wi . Train-
ing is iterative, and proceeds as a discrete set of steps where training data is
presented to the network, then the weight vectors are updated until some pre-
determined condition is met.
In the standard SOM the magnitude of the weight update is dependent on
the iteration number or time. For the PLSOM [2] and PLSOM2 [1] algorithms,
on the other hand, the weight update is a function of what can be inferred about
the state of the network.
3 Learning Algorithm
Recursion in SOM type neural networks works the same way as recurrence in
other Neural Network algorithms: The output at time t 1 forms part of the
input at time t. The particular method used here is adopted from [12].
The output is dened as a vector of length n, where n is the number of nodes,
that corresponds to the excitation of a given node.
ki = e[||x(t1)wx || + (1)||y(t1)wy ||] (1)
In (1), is a tuning parameter that determines how much inuence the new
input has relative to the recurrent connections, k is a vector of length n, e is
Eulers constant and y is the output vector, wy is the part of the weight vector
that corresponds to the output (that is, the recurrent weights) and wx is the part
of the weight vector that corresponds to the input. Following the calculation of
k, the updated value of y is found by scaling and translating k so that each
element lies in the range [0, 1] according to (2).
ki min(k)
yi (t) = (2)
max(k) min(k)
Towards a Universal Modeller of Chaotic Systems 309
In other words, y is a vector where each entry yi corresponds to the excitation

of a node i. The entry that corresponds to the winning node, yc , is equal to 1,
all other entries are in the interval [0, 1. The vector y is the excitation vector.
To get the Recursive PLSOM2 algorithm (RPLSOM2), is changed to (3).
||x wx,c || + (1 )||y wy,c ||
= (3)
Sx + (1 )Sy
The part of the weight vector of the winning node that relates to the input
is wx,c , and the part that relates to the recurrent connections is wy,c . The scalar
Sy is the diameter of the set of all recurrent inputs, and Sx is the diameter of
the set of all direct inputs. The parameter is in the range [0, 1]. For = 1 the
RPLSOM2 is equal to the PLSOM2, since all recurrent connections are ignored.
3.1 Idle Mode

As is clear from the above algorithm the input is only partially responsible for
selecting the winning node, a large part is also due to the excitation vector given
by (2). The map can thus be iterated without taking the input into account by
calculating the winning node and the next value of the excitation vector based on
the current excitation vector. This will henceforth be referred to as idle mode.
4 Experimental Setup
As training input for time series are used:
1. A sine wave with wavelength 200.
2. The sum of two sine waves, with wavelengths 200 and 61.5.
3. A simulated Mackey-Glass [9] sequence, consisting of 511 sine waves with
dierent wavelength and amplitude added together to resemble the frequency
response of the real Mackey-Glass sequence.
4. A chaotic Mackey-Glass series.
All time series were scaled and translated to span the interval [0, 1]. The Fourier
coecients of of the simulated Mackey-Glass series is indistinguishable from the
Fourier coecients of the real Mackey-Glass series, as the simulated series was
created through inverse Fourier transform of the real series.
The Mackey-Glass time series is given by (4).
dx x
=a bx (4)
dt 1 + xn
here x represents the value of the variable at time index (t ). In the present
work the following values are used: = 17, a = 0.2, b = 0.1, and n = 10.
Before each test the RPLSOM2 weights were initialised to a random state
and trained with 50000 samples from one of the time series. The map has 100
nodes arranged in a 10 10 grid. The generalisation factor was set to 17, and
set to 0.6. Each experiment was repeated 1000 times for each time series.
310 E. Berglund
4.1 Repetition
One characteristic of chaotic systems is that they have very long repetition peri-
ods. Therefore the repetition period for the input sequences was measured. The
repetition period is dened as the number of samples one must draw from the
series before the last 100 samples are repeated anywhere in any of the previous
samples.
After training the map was put into idle mode for 200000 iterations. The
weight vector of the winning node will describe a one-dimensional time series,
which was checked for repetition.
4.2 Fractal Dimension
To calculate the fractal dimension of the series attractor, it was embedded in 3

dimensions, using (5),
XtT = {x(t), x(t 1), x(t 2)} (5)
where x(t) is the sample drawn from the time series at time t. This results in a
trajectory of vectors in 3-space.
The fractal dimension was estimated using the information dimension based
on 15000 samples drawn from the trained RPLSOM2 in idle mode. Any output
sequence with fewer than 100 unique points were discarded, since this gives the
information dimension computation too little to work with to give a meaningful
result. Few unique points indicate a stable orbit or stable point, which would
indicate a non-fractal dimension. The number of discarded output sequences for
each input time series are given by Table 1.
Table 1. Number of output sequences discarded because of too few unique points.
Sequence Number discarded

Sine 920
Double sine 764
Simulated Mackey-Glass 718
Mackey-Glass 160
4.3 Lyapunov Exponent
The Lyapunov exponent was calculated using the excitation vector p from (2) of
the RPLSOM2 and the numerical algorithm described in [10]. The perturbation
value used was d0 = 1012 , and the algorithm was in idle mode for 300 iterations
before 15000 iterations were sampled and averaged to compute the Lyapunov
exponent estimate.
5 Results
As can be seen from Table 2, the mean Lyapunov exponent for the map trained
with a chaotic time series is clearly less negative. This becomes even clearer from
Table 3, which shows the percentage of maps with positive Lyapunov exponents.
Table 2. Estimated Lyapunov exponent.
Sequence Mean Lyapunov exponent std err

Sine 0.2451 0.0054
Double sine 0.2326 0.0066
Simulated Mackey-Glass 0.1641 0.0035
Mackey-Glass 0.0636 0.0041
Table 3. Percentage of trained maps with a positive Lyapunov exponent
Sequence % positive exponents std err

Sine 2.3 0.5
Double sine 4.1 0.6
The connection between the chaos of the input sequence and the behaviour
of the map is also evident in the number of iterations before repeat, see Table 4.
Table 4. Mean number of iterations before map starts repeating
Sequence iterations std err

Sine 446 35
Double sine 997 57
Simulated Mackey-Glass 832 37
Mackey-Glass 2809 81
As shown Table 5 there is a clear dierence in the dimension of the dierent

maps, and the Mackey-Glass, as expected, has the largest dimension value. The
sine-trained sequence has a surprisingly high dimension, even though it is the
simplest sequence. It is possibly an artefact of the high number of tests that had
to be discarded due to too few unique points, see Table 1.
312 E. Berglund
Table 5. Mean information dimension of map output
Sequence dimension std err

Sine 1.0181 0.0024
Double sine 1.0145 0.0010
6 Conclusion
It was observed that RPLSOM2 networks trained with a chaotic time series to
a signicant degree exhibit the following characteristics:
Longer repetition periods.
Higher Lyapunov exponent.
Higher probability of having a positive Lyapunov exponent.
Higher fractal dimension.
when compared to networks that have been trained on non-chaotic but otherwise
similar time sequences. This is consistent with chaotic behaviour.
This is the rst instance of the chaotic behaviour of a network output after
training depends on its training input.
References
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445 (1993)
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Y. (eds.) Temporal Coding in the Brain. NEUROSCIENCE. Springer, Heidelberg
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ral sequence processing. In: Gerstner, W., Germond, A., Hasler, M., Nicoud, J.-D.
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12. Voegtlin, T.: Recursive self-organizing maps. Neural Netw. 15(89), 979991 (2002)
13. Wang, L., Li, S., Tian, F., Fu, X.: A noisy chaotic neural network for solving
combinatorial optimization problems: stochastic chaotic simulated annealing. IEEE
Trans. Syst. Man Cybern. Part B 34(5), 21192125 (2004)
An Approach for Generating Test Problems
of Constrained Global Optimization
Victor Gergel(B)

gergel@unn.ru
Abstract. In the present paper, a novel approach to the constructing

of the test global optimization problems with non-convex constraints is
considered. The proposed approach is featured by a capability to con-
struct the sets of such problems for carrying out multiple computational
experiments in order to obtain a reliable evaluation of the eciency of the
optimization algorithms. When generating the test problems, the neces-
sary number of constraints and desired fraction of the feasible domain
relative to the whole global search domain can be specied. The loca-
tions of the global minimizers in the generated problems are known a
priori that simplies the evaluation of the results of the computational
experiments essentially. A demonstration of the developed approach in
the application to well-known index method for solving complex multiex-
tremal optimization problems with non-convex constraints is presented.

convex constraints Test optimization problems Numerical experiments
1 Introduction
In the present paper, the methods for generating the global optimization test
problems with non-convex constraints
(y ) = min {(y) : y D, gi (y) 0, 1 i m}, (1)

D = y RN : ai yi bi , 1 i N (2)
are considered. The objective function (y) (henceforth denoted by gm + 1 (y))

and the left-hand sides gi (y), 1 i m, of the constraints are supposed to
satisfy the Lipschitz condition
|gi (y ) gi (y )| Li y y , y , y D, 1 i m + 1.
with the Lipschitz constants unknown a priori. The analytical formulae of the
problem functions may be unknown, i.e. these ones may be dened by an
algorithm for computing the function values in the search domain (so called
black-box-functions). It is supposed that even a single computing of a prob-
lem function value may be a time-consuming operation since it is related to the
necessity of numerical modeling in the applied problems (see, for example, [14]).
https://doi.org/10.1007/978-3-319-69404-7_24
An Approach for Generating Test Problems 315
The evaluation of eciency of the developed methods is one of the key prob-
lems in the optimization theory and applications. Unfortunately, it is dicult
to obtain any theoretical estimates in many cases. As a result, the comparison
of the methods is performed by carrying out the computational experiments on
solving some test optimization problems in most cases. In order to obtain a reli-
able evaluation of the eciency of the methods, the sets of test problems should
be diverse and representative enough. The problem of choice of the test problems
has been considered in a lot of works (see, for example, [58]). Unfortunately, in
many cases, the proposed sets contain a small number of test problems, and it
is dicult to obtain the problems with desired properties. The most important
drawback consists of the fact that the constraints are absent in the proposed
test problems as a rule (or the constraints are relatively simple: linear, convex,
etc.).
A novel approach to the generation of any number of the global optimiza-
tion problems with non-convex constraints for performing multiple computa-
tional experiments in order to obtain a reliable evaluation of the eciency of the
developed optimization algorithms has been proposed. When generating the test
problems, the necessary number of constraints and desired fraction of the feasi-
ble domain relative to the whole search domain can be specied. In addition, the
locations of the global minimizers in the generated problems are known a priori
that simplies the evaluation of the results of the computational experiments
essentially.
2 Test Problem Classes

A well-known approach to investigating and comparing the multiextremal opti-
mization algorithms is based on testing these methods by solving a set of prob-
lems, chosen randomly from some specially designed class.
One generator for random samples of two-dimensional test functions has been
described in [9,10]. This generator produces two-dimensional functions according
to the formula
7 2
(y) = 7
i=1 j = 1 Aij gij (y) + Bij hij (y) +
7 7
2 1/2
i=1 j = 1 Cij gij (y) + Dij hij (y) , (3)
where gij (y) = sin(iy1 ) sin(jy2 ), hij (y) = cos(iy1 ) cos(jy2 ), y = (y1 , y2 )
R2 , 0 y1 , y2 1, and coecients Aij , Bij , Cij , Dij are taken uniformly in the
interval [1, 1].
Let us consider a scheme for constructing the generator GCGen (Global
Constrained optimization problem Generator) which allows to generate the test
global optimization problems with m constraints. Obviously, one can generate
m + 1 functions, the rst m of these ones can be considered as the constraints
and the (m + 1)-th function as the objective one. However, in this case, the
conditional global minimizer of the objective function is unknown, and the pre-
liminary estimate of this one (for example, by scanning over a uniform grid)
316 V. Gergel
will be time-consuming. At the same time, one could not control the size of the
feasible domain. In particular, the constraints might be incompatible, and the
feasible domain might be empty.
Below, the rules, which allow formulating the constrained global optimization
problems so that:
one could control the size of feasible domain with respect to the whole domain
of the parameters variation;
the global minimizer of the objective function would be known a priori taking
into account the constraints;
the global minimizer of the objective function without accounting for the
constraints would be out of the feasible domain (with the purpose of simulat-
ing the behavior of the constraints and the objective function in the applied
constrained optimization problems)
are proposed.
The rules dening the operation of the generator of the constrained global
optimization with the properties listed above consist in the following.
1. Let us generate m + 1 functions fj (y), y D, 1 j m + 1, by some
generating scheme (for instance, by using the formula (3)). The constraints
will be constructed on the base of the rst m functions, the (m+1)-th function
will serve for the construction of the objective function.
2. In order to know the global minimizer in the constrained problem a priori,
let us make it to be the same to the global minimizer in the unconstrained
problem. To do so, let us perform a linear transformation of coordinates
so that the global minimizers of the constraint functions yj , 1 j m,

would transit into the minimizer of the objective function ym + 1 . This way,
the functions fj (y), 1 j m, with the same point of extremum will be
constructed.
3. In order to control the size of the feasible domain, let us construct an auxiliary
function (a combined constraint)
H(y) = max fj (y)

1jm
and compute its values in the nodes of a uniform grid in the domain D;
the number of the grid nodes in the conducted
experiments
should be big
enough (in our experiments it was min 107 , 102N ). Then, let us nd the
maximum and minimum values of the function H(y) in the grid nodes, Hmax
and Hmin , respectively, and construct a characteristic s(i) the number of
points, in which the values of H(y) fall into the range

Hmax Hmin
Hmin , Hmin + i , 1 i 100.
100
Then, the functions
Hmax Hmin
fj (y) q = Hmin + i , 1 j m,
100
where i is selected to be the minimal one satisfying the inequality

s(i)

s(100)
will construct a problem with the feasible domain occupying the fraction
, 0 < < 1, of the whole search domain.
4. The test problem of global constrained optimization can be stated as follows

min (y) : y D, gj (y) = fj (y) q 0, 1 j m
where

m

(y) = fm + 1 (y) max 0, fj (y) q ,
j =1
where is a positive integer number and > 0 is selected in such a way as

to provide the global minimum location of the function (y) in the infeasible
part of the search domain D. For instance, to guarantee this property the
value of can be set as follows
> (hmax hmin )/(Hmax q) ,
where hmax and hmin are the maximum and minimum values of the function
fm + 1 (y) respectively.
3 Some Numerical Results

As an illustration, the level lines of the objective functions and the zero-level
lines of the constraints for problems constructed on the base of functions (3)
with = 3, = 1 and the volume fractions of the feasible domains = 0.4, 0.6
are shown in Fig. 1. The feasible domains are highlighted by green. The change
of volume and, at the same time, the increase of complexity of the feasible
domains are seen clearly. Figure 1(a,b) also shows the points of 628 and 764
trials, correspondingly, performed by the index method for solving constrained
global optimization problems until the required accuracy = 102 was achieved.
The conditional optimizer is shown as a red point and the best estimation of the
optimizer is shown as a blue point.
The index method has been proposed and developed in [1113]. The approach
is based on a separate accounting for each constraint of the problem and is
not related to the use of the penalty functions. According to the rules of the
index method, every iteration includes a sequential checking of fulllment of
the problem constraints at this point. The rst occurrence of violation of any
constraint terminates the trial and initiates the transition to the next iteration.
This allows: (i) accounting for the information on each constraint separately and
(ii) solving the problems, in which the function values may be undened out of
the feasible domain. It should be noted, that the index method can be eciently
parallelized for accelerators [14,15].
318 V. Gergel
(a) (b)
Fig. 1. The problems based on functions (3)
4 Conclusion
This paper considers a method for generating global optimization test problems
with non-convex constraints that allows:
to control the size of feasible domain with respect to the whole domain of the
parameters variation;
to know a priori the conditional global minimizer of the objective function;
to generate the unconditional global minimizer of the objective function out
of the feasible domain (to simulate the constraints and objective function in
the applied optimization problems).
The demonstration of the developed approach in application to well-known

index method for solving complex multiextremal optimization problems with
non-convex constraints is considered.
The developed approach allows generating any number of test global opti-
mization problems with non-convex constraints for performing multiple compu-
tational experiments in order to obtain a reliable evaluation of the eciency
of the developed optimization algorithms. To develop the proposed approach
further, the development of new test classes for the optimization problems of
various dimensionalities is planned.

project No 16-11-10150.
References
algorithms based on ecient domain partitions. Electr. Power Syst. Res. 78(7),
12171229 (2008)
3. Kvasov, D.E., Sergeyev, Y.D.: Deterministic approaches for solving practical black-
box global optimization problems. Adv. Eng. Softw. 80, 5866 (2015)
in design of scientic products for purposes of cavitation problems. Solving global
optimization problems on GPU cluster. In: Simos, T.E. (ed.) ICNAAM 2015, AIP
Conference Proceedings, 1738, art. no. 400013 (2016)
tion. Kluwer Academic Publishers, Dordrecht (1999)
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Glob. Optim. 31(4), 635672 (2005)
8. Addis, B., Locatelli, M.: A new class of test functions for global optimization. J.
Glob. Optim. 38(3), 479501 (2007)
Stat. Optim. 7, 198206 (1978). [in Russian]
straints. Sequential and Parallel Algorithms. Kluwer Academic Publishers,
Dordrecht (2000)
12. Sergeyev, Y.D., Famularo, D., Pugliese, P.: Index branch-and-bound algorithm for
Lipschitz univariate global optimization with multiextremal constraints. J. Glob.
Optim. 21(3), 317341 (2001)
(2002)
14. Barkalov, K., Gergel, V., Lebedev, I.: Use of Xeon Phi coprocessor for solving global
15. Barkalov, K., Gergel, V.: Parallel global optimization on GPU. J. Glob. Optim.
66(1), 320 (2016)
Global Optimization Using Numerical
Approximations of Derivatives
Victor Gergel and Alexey Goryachih(B)
Lobachevsky State University of Nizhni Novgorod,

Gagarin Prospect, 23, Nizhni Novgorod 603600, Russian Federation
gergel@unn.ru, a goryachih@mail.ru
http://www.unn.ru
Abstract. This paper presents an ecient method for solving global

optimization problems. The new method unlike previous methods, was
developed, based on numerical estimations of derivative values. The
eect of using numerical estimations of derivative values was studied
and the results of computational experiments prove the potential of such
approach.
Keywords: Multiextremal optimization Global search algorithm

Lipschitz condition Numerical derivatives Computational experiments
1 Introduction
The global optimization problem is a problem of nding the minimum value of
a function (x)
(x ) = min{(x) : x [a, b]}. (1)
For numerical solving of the problem (1) optimization methods usually gen-
erate a sequence of points yk which converges to the global optimum x .
Suppose that the optimized function (x) is multiextremal. Also assume that
the optimized function (x) satises the Lipschitz condition
|(x2 ) (x1 )| L |x2 x1 | , x1 , x2 [a, b], (2)
where L > 0 is the Lipschitz constant. In addition to (2) also assume that the

rst derivative of the optimized function (x) satises the Lipschitz condition

(x2 ) (x1 ) L1 || x2 x1 || , x1 , x2 [a, b]. (3)
2 One-Dimensional Global Optimization Algorithm

Using Numerical Estimations of Derivatives
Conditions (2) and (3) allow estimating possible values of the optimized func-
tion (x) more accurately, improving eciency of global optimization methods.
This paper presents an improvement of global optimization algorithms that uses
derivatives of the optimized function (x).
https://doi.org/10.1007/978-3-319-69404-7_25
Global Optimization Using Numerical Approximations of Derivatives 321
2.1 Core One-Dimensional Global Search Algorithm

Using Derivatives
The new method that is proposed in this paper based on the Adaptive Global
Method with Derivatives (AGMD). This method was developed in the frame-
work of information-statistical theory of multiextremal optimization [4,5]. This
method was proposed in [1] and a similar approach was described in [3].
The computational scheme of the AGMD can be given as follows.
The rst two iterations are executed at the boundary points of the interval
[a, b]. Assume that k ( k > 1) iterations of global search have been performed,
each iteration includes a calculation of a value of the optimized function (x)

and its derivative (x), this calculation will be further referred to as trial.
Next point of trial of (k + 1) iteration is calculated according to the following
rules:
Rule 1. Renumber the points of previous trials by subscripts in increasing order
a = x0 < x1 < < xi < < xk = b.
Rule 2. Compute the estimation of the Lipschitz constant from (3) for the rst
derivative of the optimized function

| (x ) (xi1 )|/|xi xi1 |,
i

Mi = max 2 (xi ) (xi1 ) (xi1 )(xi xi1 ) /(xi xi1 ) ,
2

2 (x ) (x ) (x )(x x ) /(x x )2 ,

i i1 i i i1 i i1
(4)
M = max(Mi ), 1 i k,

rM, M > 0,
m=
1, M = 0,
where r > 1 is the reliability parameter of the algorithm.
Rule 3. Construct a minorant for the optimized function on each interval
(xi , xi+1 ), 0 i k 1:

2

(xi ) + (xi )(x xi ) 0.5 m(x xi ) , x (xi , xi ),
min
2
(x) = Ai (x xi ) + 0.5 m(x x i ) + Bi , x [xi , xi ],

(xi+1 ) + (xi+1 )(x xi+1 ) 0.5 m(x xi+1 )2 , x (xi , xi+1 ),
(5)
in order to make the minorant and its rst derivative continuous the intervals

(xi , xi ) and the values Ai and Bi are dened by the relation
[(xi ) (xi+1 )xi+1 ] + m(x2i+1 x2i )/2 md2x
xi = ,
m(xi+1 xi ) + ( (xi+1 ) (xi ))
[(xi ) (xi+1 )xi+1 ] + m(x2i+1 x2i )/2 + md2x
xi = ,
m(xi+1 xi ) + ( (xi+1 ) (xi ))

where the intervals (xi , xi ) and the values Ai and Bi are dened in order to
make the minorant and its rst derivative continuous (see [1]).
322 V. Gergel and A. Goryachih
Rule 4. For each interval (xi , xi+1 ) (0 i k 1) the characteristic

min (xmin
i ), xmin
i [xi , xi ],
R(i) =
min(min (x i ), min (xi )), xmin
i
/ [xi , xi ],
where
(xi1 ) + m(xi xi1 ) + mxi
xmin
i = .
m
Rule 5. Find the interval (xt1 , xt ) with the minimal characteristic R(t)
R(t) = min {R(i) : 1 i k} . (6)
In the case when there exist several intervals satisfying (6) the interval with
the minimal number t is taken for certainty.
Rule 6. Compute the next point of the next trial xk+1 accordingly

min min
xt , xt [xt , xt ],

xk+1 = xt , min (x t ) min (xt ),

xt , min (x t ) > min (xt ).
The stopping condition is dened by the following relation
|xt xt1 | , (7)
where is the accuracy, > 0.
Convergence conditions of the algorithm are described in [1].
2.2 One-Dimensional Global Search Algorithm Using Numerical

Derivatives
As it was mentioned above, the derivative may be unknown or its values are
time-consuming to compute. In this paper the modication of the AGMD based
on numerical dierentiation is proposed.
The following relations are used for numerical estimations of values of the
rst derivative:
left-hand and right-hand approximation of values of the rst derivatives by

two values of the function
(xi+1 ) (xi )
i = , 0 i k 1,
xi+1 xi
(xk ) (xk1 )
i = ,
xk xk1
approximation of values of the rst derivatives at left, right and center points
by three values of the function

1 (1 + 2 )2 1
0 = 2 (2 + 2 )(x0 ) + (x1 ) (x2 ) ,
H1 2 2

1 (1 + k )2 (2 + k )
k = k k (xk2 ) (xk1 ) + (xk ) ,
Hk1 k k
for 1 i k 1

1 2
(i+1 1) 1
i = i+1 (xi1 ) (xi ) + (xi+1 ) ,
Hii+1 i+1 i+1
where Hii+1 = hi + hi+1 , i+1 = hhi+1i

and hi = xi xi1 . Additional details can
be found in [11].
The algorithm of AGMD with numerical estimations of values of the rst
derivative will be further referred to as the Adaptive Global Method with Numer-

ical Derivatives (AGMND). In the scheme of AGMND we just replace (xi ) by

i (0 i k), it forms two modications of AGMND: AGMND-2 and AGMND-
3 with approximation of derivative values by 2 or 3 values of the optimized
functions correspondingly.
3 Results of Computational Experiments

For the rst series of experiments ve well known methods of global optimization
were compared: Galperins algorithm (GA), Piyavskiis algorithm (PA), Global
Search Algorithm (GSA) proposed by Strongin, AGMD and AGMND with three
dierent scheme of approximation of derivative values by 2 and 3 values of the
function AGMND-2, AGMND-3 correspondingly.
Within the executed experiments a set of multiextremal functions from [1]
(it is also used in [2]) was chosen.
The accuracy of global search is equal to = 104 (ba). Galperins algorithm
and Piyavskiis algorithm use the exact Lipschitz constant. For GSA parameter
of methods r is equal to 2. For AGMD and AGMND (AGMND-2, AGMND-3)
parameter of methods r is equal to 1.1. All results (except AGMND) were taken
from [1].
Table 1 contains the results of methods in term of the numbers of trials that
had to be performed before the stopping condition (7) (empty cells mean that
exact solutions were not found). The last row in the table contains the average
number of trials from nonempty cells.
As shown in the results of AGMND-3 and AGMD are quite close. It is impor-
tant to note that the method with numerical derivatives looks even more eective
since each trial in AGMD includes calculation of the function and its derivative
in the relations (4) and (5) from the scheme. But in some cases AGMND-3 did
not nd the global minimum before the stopping condition (7) was satised.
324 V. Gergel and A. Goryachih
Table 1. The results of comparison of one-dimensional methods of global optimization
GA PA GSA AGMD AGMND-2 AGMND-3

1 377 149 127 16 25 17
2 308 155 135 13 30 14
3 581 195 224 50 136 67
4 923 413 379 15 27 12
5 326 151 126 14 40 18
6 263 129 112 22 51 28
7 383 153 115 13 32 14
8 530 185 188 47 147 42
9 314 119 125 12 33 16
10 416 203 157 12 30 -
11 779 373 405 29 58 29
12 746 327 271 23 48 16
13 1829 993 472 59 26 20
14 290 145 108 15 48 16
15 1613 629 471 41 71 63
16 992 497 557 49 27 25
17 1412 549 470 44 90 42
18 620 303 243 10 32 11
19 302 131 117 12 32 15
20 1412 493 81 24 51 24
Average 720.8 314.6 244.15 26 51.7 24.83
In the next experiment dierent values of the reliability parameter r were

compared with the following dynamic schemes for setting r: r = r + d/k,
where k was a number of trials, r = 1.1, d = 10.
Results of the experiment with the constant parameters r = 1.1, 1.5, 2.0 and
with variable parameters for AGMND-3 are presented in Table 2. These results
show that the proposed dynamic scheme are the most ecient.
Table 2. The results of comparison of dierent schemes for setting the reliability
parameter
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Average
1.0 17 14 67 12 18 28 14 43 16 - 29 16 20 16 63 25 - 11 15 24 24.89
1.5 21 15 66 16 22 27 16 64 20 - 28 25 2 17 38 32 - 14 17 28 27.17
2.0 23 18 77 20 24 28 20 72 22 21 30 30 29 28 35 38 47 16 18 28 31.2
r + d/k 18 14 49 16 18 24 17 47 16 16 30 21 25 17 34 30 32 15 15 25 23.95
4 Conclusion
In the framework of the proposed approach to solving global optimization prob-
lems, the algorithm of AGMND-3 showed results close to results of AGMD. The
method with numerical derivatives looks even more eective since each trial in
AGMD includes calculation of the function and its derivative.
For further research it is necessary to continue computational experiments on
higher-dimensional optimization problems, as well as to provide some theoretical
basis for AGMND.
Acknowledgments. This research was supported by the Russian Science Foundation,

project No 16-11-10150 Novel ecient methods and software tools for time-consuming
decision making problems using supercomputers of superior performance.
References
1. Gergel, V.P.: A method of using derivatives in the minimization of multiextremum
functions. Comput. Math. Math. Phys. 36(6), 729742 (1996)
2. Sergeyev, Y.D., Mukhametzhanov, M.S., Kvasov, D.E., Lera, D.: Derivative-free
local tuning and local improvement techniques embedded in the univariate global
optimization. J. Optim. Theor. Appl. 171(1), 186208 (2016)
3. Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary func-
tions. Math. Program. 81(1), 127146 (1998)
straints: Sequential and Parallel Algorithms. Kluwer Academic Publishers,
Dordrecht (2000)
5. Strongin, R.G.: Numerical methods in multiextremal problems: information-
statistical algorithms. Nauka, Moscow (1978). (in Russian)
6. Barkalov, K., Gergel, V.P.: Parallel global optimization on GPU. J. Global Optim.
66(1), 320 (2016)
7. Gergel, V.P., Kuzmin, M.I., Solovyov, N.A., Grishagin, V.A.: Recognition of surface
defects of cold-rolling sheets based on method of localities. Int. Rev. Automat.
Control 8(1), 5155 (2015)
8. Barkalov, K., Gergel, V., Lebedev, I.: Use of xeon phi coprocessor for solving global
9. Paulavicius, R., Zilinskas, J.: Advantages of simplicial partitioning for Lipschitz
optimization problems with linear constraints. Optim. Lett. 10(2), 237246 (2016)
10. Paulavicius, R., Sergeyev, Y.D., Kvasov, D.E., Zilinskas, J.: Globally-biased DIS-
IMPL algorithm for expensive global optimization. J. Global Optim. 59(23), 545
567 (2014)
11. Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of
Algorithmic Dierentiation, 2nd edn. Society for Industrial and Applied Mathe-
matics, Philadelphia (2008)
Global Optimization Challenges in Structured
Low Rank Approximation
Jonathan Gillard1(B) and Anatoly Zhigljavsky1,2

1
School of Mathematics, Cardi University, Cardi CF24 4AG, UK
{gillardjw,zhigljavskyaa}@cardiff.ac.uk
2
Lobachevsky Nizhny Novgorod State University,
23 Prospekt Gagarina, 603950 Nizhny Novgorod, Russia
Abstract. In this paper, we investigate the complexity of the numerical

construction of the so-called Hankel structured low-rank approximation
(HSLRA). Briey, HSLRA is the problem of nding a rank r approxi-
mation of a given Hankel matrix, which is also of Hankel structure.
Keywords: Structured low rank approximation Hankel matrices

Time series analysis
1 Statement of the Problem

The aim of low-rank approximation methods is to approximate a matrix con-
taining observed data, by a matrix of pre-specied lower rank r. The rank of
the matrix containing the original data can be viewed as the order of complex-
ity required to t to the data exactly, and a matrix of lower complexity (lower
rank) close to the original matrix is often required. A further requirement is
that if the original matrix of the observed data is of a particular structure, then
the approximation should also have this structure. Let L, K and r be given
positive integers such that 1 r < L K. Denote the set of all real-valued
L K matrices by RLK . Let Mr = MLK r RLK be the subset of RLK
containing all matrices with rank r, and H = HLK RLK be the subset of
RLK containing matrices of some known structure. The set of structured LK
matrices of rank r is A = Mr H. Assume we are given a matrix X H.
The problem of structured low rank approximation (SLRA) is:
f (X) min (1)
XA
where f (X) = d(X, X ) is a squared distance on RLK RLK . In this paper

we only consider the case where H is the set of Hankel matrices and thus refer
to (1) as HSLRA.
One distance to dene f in (1) is given by
||X||2Q,R = Trace QXRXT , (2)
where Q and R are positive denite matrices. We will call this norm the (Q, R)-
norm. Note that if Q and R are identity matrices then (2) denes the Frobenius
norm (which is the most popular choice).
https://doi.org/10.1007/978-3-319-69404-7_26
Optimization Challenges in SLRA 327
2 Optimization Challenges
2.1 Challenge 1: Selecting f
The selection of the distance f represents one of the challenges in HSLRA. To

motivate this challenge, consider a problem of modelling a time series. Assume
we are given a vector Y = (y1 , . . . , yN )T RN . A common problem of time
series analysis is
(S, Y ) min (3)
SL r
where (, ) is a distance on RN RN and Lr RN is the set of vectors in

RN which satisfy a linear recurrence relation (LRR) of order r; we say that a
vector S = (s1 , . . . , sN )T satises an LRR of order r if
sn = a1 sn1 + . . . + ar snr , for all n = r + 1, . . . , N, (4)
where a1 , . . . , ar are some real numbers with ar = 0. The model (4) includes, as
a special case, the model of a sum of exponentially damped sinusoids:
q

sn = a exp (d n) sin (2 n + ), n = 1, . . . , N, (5)
=1
where typically q = r/2.

The optimization problem (3) can be equivalently formulated as a matrix
optimization problem, where vectors in (3) are represented by L K Hankel
matrices. With a vector Z = (z1 , . . . , zN )T of size N and given L < N we
associate an L K Hankel matrix

z1 z2 zK
z2 z3 zK+1

XZ = . .. .. .. H ,
.. . . .
zL zL+1 zN
where K = N L + 1. We also write this matrix as XZ = H(Z) and note that

H makes a one-to-one correspondence between the spaces RN +1 and H so that
for any matrix X H we may uniquely dene Z = H1 (X) with X = H(Z).
The matrix version of the optimization problem (3) can now be written as
d(X, XY ) min (6)

XA
where d(, ) is a distance on RLK RLK and A is the set of all L K Hankel
matrices of rank r. This is the optimization problem (1).
The optimization problems (3) and (1) are equivalent if the distance functions
(, ) in (3) and d(, ) in (1) are such that
(Z, Z ) = c d(H(Z), H(Z )) (7)

328 J. Gillard and A. Zhigljavsky
for Z, Z RN , where c > 0 is arbitrary. It can be assumed that c = 1 with-

out loss of generality. The standard choice of the distance d(, ) in the HSLRA
problem (1) is d(X, X ) = X X F , where F is the Frobenius norm. The
primary reason for this choice is the availability of the singular value decomposi-
tion (SVD) which constitutes the essential part of many algorithms attempting
to solve the HSLRA problem (1).
However, if the distance d(X, X ) in (1) is d(X, X ) = X X F then the
distance in (3) takes a particular form. One would prefer to dene the distance
function (, ) in (3) and acquire the distance d(, ) for (1) from (7), rather
than vice-versa, which is a common practice. Dierent distances (, ) can be
used and may be desired. There is one serious problem, however, related to
the complexity of the resulting HSLRA problem (1). If d(, ) is dened by the
Frobenius norm then the HSLRA problem (1), despite being dicult, is still
considered as solvable since there is a very special tool available at intermediate
stages, the SVD. If d(, ) does not allow the use of SVD or similar tools then
the HSLRA problem (1) becomes practically unsolvable (except, of course, for
some very simple cases).
Work in the paper [6] extends the choice of the norms that dene distances
in (1) and (3) preserving the availability of the SVD. These norms allow the
construction of exactly the same algorithms that can be constructed for the
Frobenius norm and, since the family of the norms considered is rather wide,
we are able to exactly or approximately match any given distance in (3). More
precisely, in [6] we consider the class of distances in (3) of the form (Z, Z ) =
Z Z W with
N
T
ZW = Z WZ =
2
wn zn2 . (8)
n=1
Further challenges arise when observations of the time series Y are classied
as exact or missing. In the former case the observation would require innite
weight, whilst in the latter the observation would require zero weight. Both
cases give rise to diculties computing with innite and innitesimals, and there
is need for methodology that allows one to represent innite and innitesimal
numbers by a nite number of symbols to execute arithmetical operations. There
is great potential for the use of grossone and the innity computer (see [11] for
more details on these topics).
2.2 Challenge 2: Complexity of the Optimization Problem

Natural approaches for solving the initial optimization problem (1) would use
global optimization techniques for optimizing parameters in either representa-
tion (4) or (5). In the case of (4), the parameters are the coecients of the
LRR: a1 , . . . , ar , and the initial values s1 , . . . , sr1 . If we were to use (5) then
the set of parameters is {(a , d , , ), = 1, . . . , q}. In both cases, the para-
metric optimization problem is extremely dicult with multi-extremality and
large Lipschitz constants of the objective functions [4]. The number of local
minima is known to increase linearly with the number of observations. Many of
Optimization Challenges in SLRA 329
the existing algorithms depend on local optimization based algorithms and do

not move signicantly from a starting point, see [2,3], and [5]. The diculty of
solving parametric versions of (3) is well understood and that is the reason why
HSLRA described by (1), the equivalent matrix formulation of (3), is almost
always considered instead of (3). As already stated, there is little work in the
literature describing how to use weights in HSLRA. However, some recent work
has commented that even unstructured weighted low-rank approximation is dif-
cult, see [7]. Increasing N leads to more erratic cost function; as N increases,
the number of local minima also increases. In many examples the number of
local minimizers increases linearly in N .
2.3 Challenge 3: Construction of Numerical Methods
One of the earliest known approaches to obtain a solution of (1) is the so-called
Cadzow iterations which are the alternating projections of the matrices, starting
at a given structured matrix, to the set of matrices of rank r (by performing
a singular value decomposition) and to the set of Hankel matrices (by diagonal
averaging). Despite the fact that Cadzow iterations guarantee convergence to
the set A, they can easily be shown to be sub-optimal (see [1]). They remain
popular due to their simplicity. One Cadzow iteration corresponds to the tech-
nique known as singular spectrum analysis, which has been an area of research
developed by the PI (see for example [1]). The main recent contributions to
nding a solution of (1) are described below.
1. Structured total least norm. Proposed by Park et al. [9], this class of methods
is aimed at rank reduction of a given Hankel matrix by 1 (that is, r = L 1).
2. Fitting a sum of damped sinusoids [10]. Methods in this category parameterize
the vector of observations as a sum of damped sinusoids and use the set of
unknown parameters as a feasible domain. This has been discussed earlier.
3. Local optimization methods starting at an existing approximation. Markovsky
and co-authors [8] have developed methodology and software to locally
improve an existing solution (or approximate solution) of (1).
In summary all of these methods suer from a number of aws [3] (i) the
rank of the matrix can only be reduced by one, (ii) they are based on local
optimizations and may not move signicantly from this initial approximation
and (iii) none have guaranteed convergence to the global optimum. Additionally
(and importantly) the focus in the literature has been on the case when the
distance f in (1) is taken to be the Frobenius norm (that is Q and R being the
identity matrices).
Acknowledgements. This work was supported by the project No. 15-11-30022

Global optimization, supercomputing computations, and applications of the Russian
Science Foundation.
330 J. Gillard and A. Zhigljavsky
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optimization problem. Informatica 22(4), 489505 (2011)
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A D.C. Programming Approach
to Fractional Problems
Tatiana Gruzdeva(B) and Alexander Strekalovsky
Matrosov Institute for System Dynamics and Control Theory of SB RAS,

Lermontov Str., 134, 664033 Irkutsk, Russia
{gruzdeva,strekal}@icc.ru
Abstract. This paper addresses a rather general fractional optimization

problem. There are two ways to reduce the original problem. The rst
one is a solution of an equation with the optimal value of an auxiliary
d.c. optimization problem with a vector parameter. The second one is to
solve the second auxiliary problem with nonlinear inequality constraints.
Both auxiliary problems turn out to be d.c. optimization problems, which
allows to apply Global Optimization Theory [11, 12] and develop two cor-
responding global search algorithms that have been tested on a number
of test problems from the recent publications.
Keywords: Fractional optimization Nonconvex problem Dierence

of two convex functions Equation with vector parameter Global search
algorithm
1 Introduction
We consider the following problem of the fractional optimization [1,10]
m
i (x)
(Pf ) f (x) := min, x S,
i=1
i (x) x
where S IRn is a convex set and i : IRn IR, i : IRn IR,
(H0 ) i (x) > 0, i (x) > 0 x S, i = 1, . . . , m.
The fractional programming problems arise in various economic applications

and real-life problems [10]. However, it is well-known that the sum-of-ratios
program is NP-complete [4]. Surveys on methods for solving this problem can
be found in [1,10], but the development of new ecient methods for a frac-
tional program still remains an important eld of research in mathematical
optimization.
In the recent two decades, we have succeeded in developing the Global Search
Theory, which perfectly ts optimization theory and proved to be rather ecient
in terms of computations [11,12]. Now we are going to apply this theory to
solving fractional programming problems.
https://doi.org/10.1007/978-3-319-69404-7_27
332 T. Gruzdeva and A. Strekalovsky
Generalizing the Dinkelbachs idea [3], we propose to attack such problems

with the help of reduction of the fractional program (Pf ) to solving an equation
with the optimal value function of a d.c. minimization problem and the vector
parameter that satises the nonnegativity assumption. To this end, we have to
use the solution to the following d.c. minimization problem, treated here as an
auxiliary one:
(DC) f (x) = g(x) h(x) min, x D,
where g(), h() are convex functions, and D is a convex set, D IRn .
Next, we also propose reduction of the sum-of-ratios problem to the prob-
lem of minimizing a linear function on the nonconvex feasible set given by d.c.
functions. In this case, we need to solve the following nonconvex problems
(DCC) f0 (x) min, x S, fi (x) = gi (x) hi (x) 0, i = 1, . . . n,

x
where gi (), hi () i = 1, . . . n, are convex functions, S IRn , f0 () is a continuous

function.
Thus, based on the solution of these two classes of d.c. optimization problems
we developed two new methods for solving a general fractional program.
The outline of the paper is as follows. In Sect. 2, instead of considering a
fractional program directly, we propose to combine solving of the corresponding
d.c. minimization problem (DC) with a search with respect to the vector para-
meter. In Sect. 3, we substantiate the reduction of the sum-of-ratios fractional
problem to the optimization problem with nonconvex constraints (DCC). Finally,
in Sect. 4, we show some comparative computational testing of two approaches
on instances with a small number of variables and terms in the sum.
2 Reduction to the D.C. Minimization Problem

Let us consider the following parametric optimization problem
m

(P ) (x, ) := [i (x) i i (x)] min, x S,
x
i=1
where = (1 , . . . , m ) IRm is the vector parameter.

Further, let us introduce the function V() m of the optimal value to Prob-

lem (P ): V() := inf {(x, ) | x S} = inf [i (x) i i (x)] : x S .
x x i=1
In addition, suppose that the following assumptions are fullled:
(a) V() > K, where K is a convex set from IRm ;
(H1 )
(b) K IRm there exists a solution z = z() to Problem (P )
In what follows, we say that the data of Problem (Pf ) satisfy the nonnega-
tivity condition, if the following inequalities hold
(H()) i (x) i i (x) 0 x S, i = 1, . . . , m.
A D.C. Programming Approach to Fractional Problems 333
Theorem 1. [5] Suppose that in Problem (Pf ) i (x) > 0, i (x) > 0
and the assumption (H1 ) is satisfied. In addition, let there exist a vector
0 = (01 , . . . , 0m ) K IRm at which the nonnegativity condition
(H(0 )) holds. Besides, suppose that in Problem (P0 ) the following equality
takes place:
m

V(0 ) : = inf [i (x) 0i i (x)] : x S = 0. (1)
x
i=1
Then, any solution z = z(0 ) to Problem (P0 ) is a solution to Problem (Pf ),

so that z Sol(P0 ) Sol(Pf ).
According to Theorem 1, in order to verify the equality (1), we should be able
m
to nd a global solution to Problem (P ) for every IR+ . Since i (), i (),
i = 1, . . . , m, are d.c. functions it can be readily seen that Problem (P ) belongs
to the class of d.c. minimization. As a consequence, in order to solve Prob-
lem (P ), we can apply the Global Search Theory [11,12].
Hence, instead of solving Problem (Pf ), we propose to combine solving of
m
Problem (P ) with a search with respect to the parameter IR+ in order to
m
nd the vector 0 IR+ such that V(0 ) = 0.
Denote i (x) := i (x) ik i (x), i = 1, . . . , m. Let [0, + ] be a segment for
varying . To choose + we should take into account that due to (H()) and
m
i (x) i (x)
(H0 ), we have i = 1, . . . , m : i fi (x) = f (x) x S,
i (x) (x)
i=1 i
so, for example, + can be chosen as i+ = fi (x0 ), i = 1, . . . , m.
Method for solving the equation V() = 0
+
Step 0. (Initialization) k = 0, v k = 0, uk = + , k = 2 [v k , uk ].
Step 1. Find a solution z(k ) to Problem (Pk ) using the global search strategy
for d.c. minimization [11,12].
Step 2. (Stopping criterion) If Vk := V(k ) = 0 and min i (z(k )) 0, then
i
STOP: z(k ) Sol(Pf ) (in virtue of Theorem 1).
Step 3. If Vk > 0, then set v k+1 = k , k+1 = 12 (uk + k ), k = k + 1 and go to
Step 1.
Step 4. If Vk < 0, then set uk+1 = k , k+1 = 12 (v k + k ), k = k + 1 and go to
Step 1.
k
Step 5. If min i (z(k )) < 0, then set ik+1 = i (z( ))
i (z(k ))
i : i (z(k )) < 0,
i
ik+1 = ik i : i (z(k )) 0. In addition, set v k+1 = 0, uk+1 = tk+1 k+1 ,
+
where tk+1 = , k = k + 1, and return to Step 1.
max i
i
Let us emphasize the fact that the algorithm for solving Problem (Pf ) of
fractional optimization consists of 3 basic stages: the (a) local and (b) global
searches in Problem (P ) with a xed vector parameter and (c) the method
for nding the vector parameter at which the optimal value of Problem (P )
is zero.
3 Reduction to the Problem with D.C. Constraints

In this section we reduce the fractional program to the optimization problem
with a nonconvex feasible set.
Proposition 1. [6] Let the pair (x , ) IRn IRm be a solution to the fol-
lowing problem:
m
i (x)
i min , x S, i , i = 1, . . . , m. (2)
i=1
(x,) i (x)
i (x )
Then = i , i = 1, . . . , m.
i (x )
Corollary 1. For any solution (x , ) IRn IRm to the problem (2), the
point x will be a solution to Problem (Pf ).
The inequality constraints in the problem (2) can be replaced by the equiv-
alent constraints i (x) i i (x) 0, i = 1, . . . , m, since i (x) > 0 x S.
This yields the following problem with m nonconvex constraints:
m
(P) f0 := i min , x S, fi := i (x) i i (x) 0, i = 1, . . . , m.
i=1 (x,)
We intend to solve this problem using the exact penalization approach for d.c.
optimization developed in [13]. Therefore, we introduce the penalized problem
(P ) (x) = f0 (x) + max{0, fi (x), i I} min, x S.
It can be readily seen that the penalized function () is a d.c. function. The the-
ory enables us to construct an algorithm which consists of two principal stages:
(a) local search, which provides an approximately critical point; (b) procedures
of escaping from critical points.
Actually, since > 0, (x) = G (x) H (x), H (x) := h0 (x) + hi (x),
iI

m
G (x) := (x)+H (x) = g0 (x)+ max hi (x); max[gi (x) + hj (x)] ,
i=1 iI jI, j=i
it is clear that G () and H () are convex functions.
Let the Lagrange multipliers, associated with the constraints and correspond-
ing to the point z k , k {1, 2, ...}, be denoted by := (1 , . . . , m ) IRm .
Global search scheme
Step 1. Using the local search method from [14], nd a critical point z k in (P).
m
Step 2. Set k := i . Choose a number : inf(G , S) sup(G , S).
i=1
Choose an initial 0 = G (z k ), k = (z k ).
Step 3. Construct a nite approximation
Rk () = {v 1 , . . . , v Nk | H (v i ) = + k , i = 1, . . . , Nk , Nk = Nk ()}
Step 4. Find a k -solution ui of the following Linearized Problem:
(P Li ) G (x)
H (v i ), x min, x S.
x
Step 5. Starting from the point ui , nd a KKT-point ui by the local search

method from [14].
Step 6. Choose the point uj : f (uj ) min{f0 (ui ), i = 1, ..., N }.
Step 7. If f0 (uj ) < f0 (z k ), then set zk+1 = uj , k = k + 1 and go to Step 2.
Step 8. Otherwise, choose a new value of and go to Step 3.
According to Corollary 1, the point z resulting from the global search strat-
egy will be a solution to the original fraction program. It should be noted that,
in contrast to the approach from Sect. 2, i will be found simultaneously with
the solution vector x.
4 Computational Simulations
Two approaches from above for solving the fractional programs (Pf ) via d.c. opti-
mization problems were successfully tested. The algorithm based on the method
for solving the equation V() = 0 from Sect. 2 (F1-algorithm) and the algorithm
based on the global search scheme from Sect. 3 (F2-algorithm) were applied to an
extended set of test examples for the various starting points. Several instances
of fractional problems from [2,79] with a small number of variables and a small
number of terms in the sum were used for computational experiments. Addition-
ally, randomly generated fractional problems with linear or quadratic functions
in the numerators and the denominators of ratios with up to 200 variables and
200 terms in the sum were successfully solved. All computational experiments
were performed on the Intel Core i7-4790K CPU 4.0 GHz. All convex auxiliary
problems (linearized problems) on the steps of F1-, F2-algorithms were solved
by the software package IBM ILOG CPLEX 12.6.2.
Table 1 presents results of some comparative computational testing of two
approaches (F1-, F2-algorithms) and employs the following designations: name
is the test example name; n is the number of variables (problems dimension);
m is the number of terms in the sum; f (x0 ) is the value of the goal function
to Problem (Pf ) at the starting point; f (z) is the value of the function at the
solution provided by the algorithms; it is the number of iterations of F1- or F2-
algorithms; T ime stands for the CPU time of computing (seconds).
Observe that one iteration of F1-algorithm and one iteration of F2-algorithm
dier in processing time and, therefore, cannot be compared. In the F1-algorithm
it denotes the number of times that we varied the parameter , while in the F2-
algorithm it stands for the number of iterations of the global search in solving
the nonconvex Problem (P ).
Computational experiments showed that solving of the fraction program
should combine the two approaches. For example, we can use the solution to
Problem (P ) to search for the parameter that reduces the optimal value
function of Problem (P ) to zero.
Table 1. Performance of two algorithms on test fractional program.
name n m f (x0 ) f (z) F1-algorithm F2-algorithm

it Time it Time
Problem [9] 2 1 0.333333 0.333333 19 0.02 1 0.01
0.750000 0.333333 18 0.02 2 0.01
Problem [8] 3 1 0.943038 0.931298 22 0.03 4 0.03
1.054217 0.931298 23 0.04 5 0.03
Problem [2] 2 2 4.293433 1.428571 213 0.16 2 9.82
1.627273 1.428571 108 0.08 3 0.07
Problem 1 [7] 2 2 3.524024 2.829684 191 0.26 3 0.02
2.829684 2.829684 19 0.05 1 0.02
Problem 2 [7] 3 3 3.000000 3.000000 18 0.04 1 0.04
3.136952 3.000000 321 0.24 7 0.11
Problem 3 [7] 3 3 3.000000 2.889069 333 0.34 2 0.05
2.904068 2.889069 259 0.26 2 0.05
5 Conclusions
In this paper, we showed how fractional programs can be solved by applying the
Global Search Theory of d.c. optimization. The methods developed were justied
and tested on an extended set of problems with linear or quadratic functions in
the numerators and denominators of the ratios.
Acknowledgments. This work has been supported by the Russian Science Founda-
tion, Project No. 15-11-20015.
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Objective Function Decomposition
in Global Optimization
Oleg V. Khamisov(B)
Melentiev Energy Systems Institute SB RAS, Lermontov, Russia

khamisov@isem.irk.ru
Abstract. In this paper we consider global optimization problems in

which objective functions are explicitly given and can be represented as
compositions of some other functions. We discuss an approach of reducing
the complexity of the objective by introducing new variables and adding
new constraints.
Keywords: Global optimization Decomposition Induced constraint

d.c. function
1 Introduction
In this paper we consider global optimization problems in which objective func-
tions are explicitly given and can be represented as compositions of some other
functions. Many practical problems can be formulated in a such form [8,9,11].
An approach similar to the described below was suggested earlier in [4,7,13]. In
[5] an equivalent approach was suggested for utility function, i.e. for the case
when objective composite function has some monotonicity properties.
2 Objective Function Decomposition and the Induced

Constraint
Consider the following mathematical programming problem
min g(x), x X, (1)
where g is a continuous composite function g(x) = F (f1 (x), . . . , fp (x)), F : Rp

R, fi : X R are continuous functions, X Rn is a compact set.
Introducing new variables yi = fi (x), i = 1, . . . , p we formulate the following
equivalent problem
min F (y), (2)
yi = fi (x), i = 1, . . . , p, x X. (3)
Assume that p < n or function F is less complicated than f . In such case we
can obtain a reduction in diculty of the initial problem providing that Eq. (3)
https://doi.org/10.1007/978-3-319-69404-7_28
Objective Function Decomposition in Global Optimization 339
are practically tractable. The latter means, for example, a possibility to solve
Eq. (3) in x for given y subject to inclusion x X by an ecient algorithm.
Solving problem (1) corresponds to nding optimal and feasible point simulta-
neously. In problem (2)(3) optimality (i.e. minimization in (2)) and feasibility
(i.e. determining x for a given y in (3)) stages are separated: they are performed
in dierent spaces. What is exactly done in the reduction of problem (1) to
problem (2)(3) and what is understood under objective function decomposi-
tion in this paper is deleting some complexity from the objective function to the
constraints, i.e. moving a part of diculty from the optimality stage the fea-
sibility stage. The motivation of a such decomposition is a desire to distribute
diculty of the initial problem between objective and constraints more or less
uniformly. It is necessary to mention that structure of the objective function is
given, we just use it. In this case we perform explicit decomposition. There are
many cases when we need to discover good (or ecient) decomposition. In the
latter case the decomposition is implicit.
In practical minimization of F in (2) it is quite often necessary to localize a
global minimum in some compact subset of Rp . Dene values
f i = min fi (x), f i = max fi (x), i = 1, . . . , p. (4)

xX xX
Instead of exact calculation of f i and f i we quite often have to use approximate

values y i f i , y i f i , i = 1, . . . , p. Let us dene the set Y Rp as the
image of X under nonlinear continuous mapping (or transformation) f : X
Y, f (x) = (f1 (x), . . . , fp (x)),
Y = {y Rp : y = f (x) for some x X}. (5)
Then the initial problem can be rewritten in the following way
min F (y), (6)
y i yi y i , i = 1, . . . , p, (7)
y Y. (8)
Since (8) is a reformulation of the feasibility stage constraint (3) the inclusion
y Y will be referred to as induced constraint.
3 Agreed Decomposition
We will say that the composite objective function g has agreed variable decom-
position
g(x) = F (f1 (x1 ), . . . , fp (xp )), (9)
where xi X i Rni , i = 1, . . . , p, X 1 . . . X p = X and n1 + . . . + np = n.
Conversely, we will say that the function g has disagreed variable decomposition
if g is still representable in the form (9) and xi Rni , ni < n, i = 1. . . . , p, but
n1 + . . . + np > n.
340 O.V. Khamisov
In the case of agreed variable decomposition we rewrite the problem (6)(8)

in the following way
min F (y), (10)
f i yi f i , i = 1, . . . , p, (11)
yi = fi (xi ), xi X i , i = 1, . . . , p. (12)
Note, that in (11) we use exact lower and upper bounds on yi since otherwise
the inclusion xi X i can be violated.
Due to variable decomposition property (12) we can use the following three-
stage approach.
I. Variable bounding stage. Determine values f i and f i through (4).
II. Optimal solution stage. Solve the problem (10)(11) and nd optimal
point y .
III. Feasibility stage. Solve the feasibility problem (12) for y = y andobtain
optimal point x for the initial problem. It easy to see that if F (y) = yi we
have the well-known separable problem.
Example 1. Consider the following three dimensional problem from [2]
g(x) = (x1 1)(x1 + 2)(x2 + 1)(x2 2)x23 ,
X = [2, 2] [2, 2] [2, 2].
Let F (y1 , y2 , y3 ) = y1 y2 y3 , f1 (x1 ) = (x1 1)(x1 + 2), f2 (x2 ) = (x2 + 1)(x2 2),
f3 (x3 ) = x23 , X i = [2, 2], i = 1, 2, 3.
I. Variable bounding stage. Each function fi is convex, so it easy to determine
f 1 = f 2 = 2.25, f 3 = 0, f 1 = f 2 = f 3 = 4.
II. Optimal solution stage. Since y 0 = 0 is feasible and F (y 0 ) = 0 then the

global minimum of F must be nonpositive. Lower bound f 3 = 0, hence in order
to minimize F variables y1 and y2 should be positive or negative simultaneously.
In the rst case the minimal value is attained at point y 1 = (2.25, 2.25, 4)
with F (y 1 ) = 25. In the second case the minimal value is attained at point y 2 =
(4, 4, 4) with F (y 2 ) = 36. Therefore, y = y 2 is the unique global minimum of
the optimal value stage.
III. Feasibility stage. Solving the feasibility problem for y = y = (4, 4, 4) we
obtain two solutions x1 = (2, 2, 2) and x2 = (2, 2, 2).
This example shows us that in some cases the problem can be solved analytically.
Example 2. Consider now the well-known Shubert function [10]
5

5
g(x1 , x2 ) = f1 (x1 ) f2 (x2 ) = i cos [(i + 1)x1 + i] i cos [(i + 1)x2 + i] ,
i=1 i=1
X = X1 X2 , X1 = X2 = [10, 10].
It is obvious to set F (y1 , y2 ) = y1 y2 .
I. Variable bounding stage. It is not dicult to nd out that
f 1 = f 2 = 12.87088549, f 1 = f 2 = 14.50800793.
To nd f i and f i , i = 1, 2 it is necessary to solve global univariate optimization

problems. At present time there exist ecient approaches for global univari-
ate optimization (see, for example, [12]), so we assume that such problems are
computationally tractable.
II. Optimal solution stage. Global minimum value of bilinear function F is
attained at two points y ,1 = (y1,1 , y2,1 ) = (12.87088549, 14.50800793) and
y ,2 = (y2,1 , y1,1 ) with F (y ,1 ) = F (y ,2 ) = 186.7309088.
III. Feasibility stage. For every i = 1, 2 each equation yj,i = fj (xj ), j = 1, 2
has three solutions in the corresponding xj . Hence, for every i = 1, 2 we have 9
solutions that gives 18 global minimum points in total.
It is worthwhile to note that in the Example 2 we rst found global minimum

value and then found all global minimum points, i.e. these two problems are
separated and their separate solution turned out to be easier than solution of
the initial global optimization problem.
4 Reducing the Induced Constraint to a d.c. Inequality

Recall [16] that a function h is called d.c. function if it can be represented as
h(x) = r(x) q(x), where functions r and q are convex. Let us introduced the
function p
2
(y) = min (yi fi (x)) . (13)
xX
i=1
It is well known that function is a d.c. function:

p
p

(y) = yi max
2
2fi (x)yi fi (x)
2
= 1 (y) 2 (y)
xX
i=1 i=1
with convex functions 1 and (implicit) 2 . Therefore, the induced constraint

can be rewritten as the d.c. inequality (y) 0. The nal reduced form of the
initial problem is the following
min F (y), (14)
(y) 0, (15)
y i yi y i , i = 1, . . . , p. (16)
Such reduction is eective when p < n (or even p n) and inner optimization
problem for calculating values of 1 can be eectively solved. The most appro-
priate example here is given by linear functions fi and small p, say, p 10.
Then the complexity is formed by nonconvex problem in p variables and it can
342 O.V. Khamisov
be solved by dierent kinds of branch and bounds methods. If F is a d.c. func-

tion then problem (14)(16) is a d.c. optimization problem and dierent d.c.
optimization methods (see [6,14,16]) can be used.
Comment. Assume that functions fi are Lipschitz continuous with constants Li ,
i = 1, . . . , p. For given y dene i (x) = yi fi (x), i = 1, . . . , p. Let a point x X
be given then
|i2 (x) i2 (x)| = |(i (x) i (x))(i (x) + i (x))|
|(fi (x) fi (x))| |2i (x)| 2|i (x)|Li x x x : i (x) i (x).

Hence, the less value |i (x)| the less is Lipschitz constant of i2 at point x, which
in this case can be taken as 2|i (x)|Li . This property can essentially improve
eciency of Lipschitz optimization methods [11,15] for solving problem (13).
Example 3. Let us consider Goldstein-Price function [2,10]

2
g(x1 , x2 ) = 1 + (x1 + x2 + 1) 19 14x1 + 3x21 14x2 + 6x1 x2 + 3x22

2
30 + (2x1 3x2 ) 18 32x1 + 12x21 + 48x2 36x1 x2 + 27x22 ,
X = [2, 2] [2, 2]. By linear transformation of variables g can be rewritten:

2
g(x1 , x2 ) = 1 + (x1 + x2 + 1) 3(x1 + x2 + 1)2 20(x1 + x2 + 1) + 36

2
30 + (2x1 3x2 ) (3(2x1 3x2 ) 16(2x1 + 3x2 ) + 18) .
Then, the introduction of new variables is obvious:
y1 = x1 + x2 + 1, y2 = 2x1 3x2 . (17)
For xed (y1 , y2 ) system (17) always has a unique solution in (x1 , x2 ). Then

F (y1 , y2 ) = 1 + y12 3y12 20y1 + 36 30 + y22 3y22 16y2 + 18 .
Bounds y 1 = 3, y 2 = 10, y 1 = 5, y 2 = 10 are easily calculated.

Function F is the product of two univariate positive functions F1 (y1 ) =
2
1 + y12 3y12 20y1 + 36 and F2 (y2 ) = 30 + (2x1 3x2 ) (3(2x1 3x2 ) 16
(2x1 + 3x2 ) + 18). This is the main advantage of the linear transformation. We
minimize each function separately (as we mentioned above univariate global opti-
mization assumed computationally tractable) and obtain global minimum point
y1 = 0, y2 = 3. Substituting y1 = 0, y2 = 3 in system (17) instead of y1 , y2 we
obtain globally optimal solution x1 = 0, x2 = 1 for the initial problem.
5 Conclusion
An objective function decomposition in global optimization was discussed. Due
to decomposition we can obtain reduction in solution diculty. The suggested
approach can be considered as a starting decomposition scheme depending on
properties of F . Types of decomposition are generated by dierent classes of
functions F . Among well-known classes we mention multiplicative functions,
sum of ratios functions and so on. Other types of function F can be used.
Acknowledgments. This work is supported by the RFBR grant number 15-07-08986.
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Projection Approach Versus Gradient Descent
for Networks Flows Assignment Problem
Alexander Yu. Krylatov(B) and Anastasiya P. Shirokolobova
Saint Petersburg State University, Saint Petersburg, Russia

a.krylatov@spbu.ru
Abstract. The paper is devoted to comparison of two methodologi-

cally dierent types of mathematical techniques for coping with net-
works ows assignment problem. Gradient descent and projection
approach are implemented to the simple network of parallel routes (there
are no common arcs for any pair of routes). Gradient descent demon-
strates zig-zagging behavior in some cases, while projection algorithm
converge quadratically in the same conditions. Methodological interpre-
tation of such phenomena is given.
Keywords: Networks ows assignment problem Projection operator

Gradient descent
1 Introduction
Huge amount of dierent practical problems are solved due to models of net-
works ows assignment. The most remarkable among them are road networks,
power grids and pipe networks [6,7]. The task it to estimate networks ows
assignment profile according to demands between all source-sink pairs. Gener-
ally, there are many source-sink pairs in a network (multicommodity networks).
In a multicommodity network ows from dierent commodities load common
arcs simultaneously and inuence on volume delays of each others.
In this paper we show that projection approach is more appropriate tech-
nique for coping with networks ows assignment problem than gradient descent.
Moreover, zig-zagging bahavior of gradient descent in the neighborhood of the
equilibrium solution is claried.
2 Networks Flows Assignment Problem

Consider a network presented by connected directed graph G = (N, A). Intro-
duce notation: N the set of sequentially numbered nodes of the graph G; A
the set of sequentially numbered arcs of the graph G; W the set of nodes
pairs (source and sink) of the graph G, w W ; K w the set of routes con-
necting source-sink pair w W ; xa the ow on arc a A, x = (. . . , xa , . . .);
ca capacity of arc a A, c = (. . . , ca , . . .); fkw the ow on route k K w ;
https://doi.org/10.1007/978-3-319-69404-7_29
346 A.Y. Krylatov and A.P. Shirokolobova
F w the ow demand between source-sink pair w W ; ta (xa ) the link per-

w
formance function (volume delay function) of arc a A; a,k indicator:

w 1, if the arc a A lies along to route k K w ;
a,k =
0, otherwise.
Networkss ow assignment problem in a link-route formulation was rst

oered by Dafermos and Sparrow [1,2]. This formulation could be expressed in
a form of the optimization program [6,8]:
xa
min ta (u)du,
x 0
aA
w w w w
subject to kK , fk 0wfor any k K , w W with denitional
w fk = F
constraints xa = wW kK w fkw a,k for any a A.
3 Simple Network of Parallel Routes
Network of parallel routes consists of two nodes (sources and sink) and n alterna-
tive arcs (routes). The demand between n source and sink is F . The demand F is
to be assigned among n routes: F = i=1 fi , fi 0, i = 1, n. Link performance
function is smooth non-decreasing function: ti C 1 (R+ ), ti (x) ti (y) 0 when
x y 0, x, y R+ , i = 1, n, where R+ non-negative orthant. Moreover, it
is believed that ti (x) 0, x 0 and ti (x)/x > 0, x > 0, i = 1, n.
From mathematical perspective, link-route and link-node formulations are
equivalent for the network of parallel routes. In such a case, networks ows
assignment problem could be expressed as follows:
n
fi

f = arg min ti (u)du, (1)
f 0
i=1
subject to
n

fi = F, (2)
i=1
fi 0 i = 1, n. (3)
According to results obtained in [4], there exists an explicit projection operator
to cope with the problem (1)(3). For the sake of convenience let us introduce
additional notations:
def def
ai (fi ) = ti (fi ) ti (fi )fi , bi (fi ) = ti (fi ), i = 1, n.
Then, the projection operator such as
f = (f )
Projection Approach for Networks Flows Assignment Problem 347
could be expressed explicitly via ai (fi ) and bi (fi ), i = 1, n:

m as (fs )
1 F+ a (f )
ms=1 b1s (fs ) i i
i (f ) = b i (f i ) s=1 bs (fs )
bi (fi ) for i m, (4)
0 for i > m,
when components f and t(f ) are indexed so that
a1 (f1 ) a2 (f2 ) . . . am (fm ) (5)
and m is dened from the condition

m
m

am (fm ) ai (fi ) am+1 (fm+1 ) ai (fi )
F < . (6)
i=1
bi (fi ) i=1
bi (fi )
Due to projection operator dened explicitly via (4)(6) the corresponding

projection algorithm was developed. At each iteration, the algorithm performed
the following three steps.
(k + 1) iteration:
1. To index mk components f k and t(f k ) so that
a1 (f1k ) a2 (f2k ) . . . amk (fm

k
k
).
2. To nd mk+1 mk (amount of non-zero components f k+1 ) from the condi-

tion
mk+1 mk+1
amk+1 (fm
k
k+1
) ai (fik ) amk+1 +1 (fm
k
k+1 +1
) ai (fik )
F < .
i=1
bi (fik ) i=1
bi (fik )
3. To compute f k+1 :
mk+1 as (fsk )
1 F + s=1 bs (fsk ) ai (fik )
fik+1 = k
mk+1 1 , i = 1, mk+1 ,
bi (fi ) s=1 bs (f k ) bi (fik )
s
fik+1 = 0, i = mk+1 , n.
4. Termination criterion
mk+1 1

ti (f k+1 ) ti+1 (f k+1 ) < .
i i+1
i=1
Note that quadratic convergence of this algorithm is proved [4].

4 Simulation Results
Zig-zagging behavior of such widely used gradient descent as Frank-Wolfe algo-
rithm became apparent in the 1970s [3,5]. That discussions were intuitive. Here
we investigate it in detail on the example of simple network of parallel routes.
Assume that volume delay functions for this network are dened as follows:
ti (fi ) = ci + di fi , i = 1, n.
Changing parameters ci and di , i = 1, n we could obtain dierent networks

of paralle routes. Lets dene ve dierent patterns of such a network according
to Table 1 with xed demand F = 20.
Table 1. Five dierent patterns of the network topology
Pattern 1 Pattern 2 Pattern 3 Pattern 4 Pattern 5

n=2 =2 n=2 =3 n=4 =3 n=4 =3 n=6 =2
i ci di i ci di i ci di i ci di i ci di
1 2 1 1 2 1 1 2 1 1 2 1 1 2 1
2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
3 1.25 2.5 3 200 200 3 200 200
4 1 3 4 1 3 4 1 3
5 300 300
6 1.25 2.5
Networks ows assignment problem could be formulated for each pattern.

Then corresponding problems could be solved. We test projection algorithm from
[4] and Frank-Wolfe algorithm on the network of parallel routes.
(k + 1) iteration of Frank-Wolfe algorithm:
1. Solve the linear programming subproblem
n

min ti fik fi ,
f
i=1
subject to (2) and (3). Let y k be its solution, and pk = y k f k the resulting
search direction.
2. Find a step length lk , which solves the problem min T (f k + lpk ) | 0 l 1 ,
where T is the objective function (1).
3. Let f k+1 = f k + lk pk and Rk+1 = {fik+1 | fik+1 > 0.1, i = 1, n} is the set of
used routes.
4. If
ti (f k+1 ) tj (f k+1 ) < ,
i j
i,jRk+1
Projection Approach for Networks Flows Assignment Problem 349
then terminate, with f k+1 as the approximate solution. Otherwise, let k :=

k + 1, and go to Step 1.
Amount of iterations required by these two algorithms are available in

Table 2. Innite amount of iterations () means zig-zagging behavior of algo-
rithm.
Table 2. Projection algorithm versus Frank-Wolfe algorithm: amount of iterations
Pattern 1 Pattern 2 Pattern 3 Pattern 4 Pattern 5

FW-algorithm 2 2 3
Projection algorithm 2 2 3 7 3
Highly remarkable that simple topology of the network allows us to draw

revealing insights. Indeed, solutions of networks ows assignment problem cor-
responding to pattern 1, 2 and 3 have no zero components, i.e. there are no
unused routes. However, solutions of networks ows assignment problem corre-
sponding to pattern 4 and 5 have zero components, i.e. there are unused routes.
Really, route 3 in pattern 4, and routes 3 and 5 in pattern 5 are obviously too
expensive to use. Thus, it is quite clear in advance that fi = 0 for i = 3 in
pattern 4 and fi = 0 for i {3, 5} in pattern 5.
According to Table 2, Frank-Wolfe algorithm demonstrates zig-zagging
behavior when there are zero components in equilibrium solution of networks
ows assignment problem. Nevertheless, projection algorithm does not pay much
attention to such a trouble (zero components in equilibrium solution) and
demonstrates high convergence rate. Projection algorithm demonstrates such a
performance primarily due to the third step of each iteration. Indeed, each iter-
ation claries zero components in equilibrium solution and excludes them from
consideration (7)-II. Eventually, projection algorithm seek solution in the space
of non-zero components (red routes on Fig. 1b corresponds to zero components).
k+1
f1k f1
f1k k+1
f2k f k+1 f1
k 2k+1 f2k
f3 ..
f3 .. .
.. . . (7)
. .. k
k+1
fm k+1
k fm
fn fnk+1 k
I II
In turn, Frank-Wolfe algorithm operate in the space of all components (7)-

I (Fig. 1a). Then it experienced zig-zagging behavior in neighborhood of zero
components. The larger the dierence between alternative routes, the higher
probability to experience zig-zagging behavior.
a) FW-algorithm b) Projection algorithm
Fig. 1. Network of parallel routes
5 Conclusion and Future Work

Obtained results show obvious advantage of projection approach for networks
ows assignment problem. The eciency of the developed projection algorithm,
rst of all, caused by excluding of zero components during solving. Therefore, it
is highly promising to develop projection algorithms. Generally, it is quite com-
plicated task to nd projection operator. Nevertheless, as it was shown in [4], for
networks ows assignment problem projection operator could be obtained in an
explicit form. Certainly, projection operator was obtained for the case of simple
network topology. However, there is a good chance to apply methodology, imple-
mented in paper [4], to a network of general topology via parallel decomposition
algorithms [6].
Acknowledgement. The rst author was jointly supported by a grant from the
Russian Science Foundation (Project No. 17-71-10069).
References
1. Dafermos, S.C., Sparrow, F.T.: The trac assignment problem for a general net-
work. J. Res. Nat. Bur. Stan. 73B, 91118 (1969)
2. Dafermos, S.-S.C.: Trac assignment and resource allocation in transportation net-
works. PhD thesis. Johns Hopkins University, Baltimore, MD (1968)
3. Holloway, C.A.: An extension of the Frank and Wolfe method of feasible directions.
Math. Program. 6, 1427 (1973)
4. Krylatov, A.Y.: Network ow assignment as a xed point problem. J. Appl. Ind.
Math. 10(2), 243256 (2016)
5. Meyer, G.G.L.: Accelerated Frank-Wolfe algorithms. SIAM J. Control 12, 655663
(1974)
6. Patriksson, M.: The Trac Assignment Problem: Models and Methods. Dover Pub-
lications, Inc., Mineola (2015)
7. Popov, I., Krylatov, A., Zakharov, V., Ivanov, D.: Competitive energy consumption
under transmission constraints in a multi-supplier power grid system. Int. J. Syst.
Sci. 48(5), 9941001 (2017)
8. She, Y.: Urban Transportation Networks: Equilibrium Analysis with Mathemati-
cal Programming Methods. Prentice-Hall, Inc., Englewood Clis (1985)
An Approximation Algorithm for Preemptive
Speed Scaling Scheduling of Parallel Jobs
with Migration
Alexander Kononov and Yulia Kovalenko(B)
Sobolev Institute of Mathematics,

4, Akad. Koptyug Avenue, 630090 Novosibirsk, Russia
alvenko@math.nsc.ru, julia.kovalenko.ya@yandex.ru
Abstract. In this paper we consider a problem of preemptive schedul-

ing rigid parallel jobs on speed scalable processors. Each job is specied
by its release date, its deadline, its processing volume and the number
of processors, which are required for execution of the job. We propose a
new strongly polynomial approximation algorithm for the energy mini-
mization problem with migration of jobs.
Keywords: Parallel jobs Speed scaling Scheduling Migration

Approximation algorithm
1 Introduction
Energy consumption of computing devices is an important issue in our days [9].
A popular technology to reduce energy usage is dynamic speed scaling, where
a processor may vary its speed dynamically. Running a job at a slower speed
is more energy ecient, however it takes longer time and may aect the per-
formance. One of the algorithmic and complexity study of this area is devoted
to revising classical scheduling problems with dynamic speed scaling (see e.g.
[1,4,6,7,9,13,14] and others).
In our paper we consider a basic speed scaling scheduling of parallel jobs.
Given a set J = {1, . . . , n} of parallel jobs to be executed on m parallel speed-
scalable processors. Each job j J is associated with a release date rj , a deadline
dj and a processing volume (work) Wj . Moreover, job j J simultaneously
requires exactly sizej processors at each time point when it is in process. Such
jobs are called rigid jobs [8].
We distinguish two variants of the problem. The rst variant (non-migratory
variant) allows the preemption of the jobs but not their migration. This means
that a job may be interrupted and resumed later on the same subset of sizej
processors, but it is not allowed to continue its execution on a dierent subset of
sizej processors. In the second variant (migratory variant) both the preemption
and the migration of jobs are allowed.
The standard homogeneous model in speed-scaling is considered. When a
processor runs at a speed s, then the rate with which the energy is consumed
https://doi.org/10.1007/978-3-319-69404-7_30
352 A. Kononov and Y. Kovalenko
(the power ) is s , where > 1 is a constant (usually, 3 [9]). Each of m

processors may operate at variable speed. However, we assume that if processors
execute the same job simultaneously then all these processors run at the same
speed. It is supposed that a continuous spectrum of processor speeds is available.
The purpose is to nd a feasible schedule, minimizing the total energy consumed
on all the processors.
The idea of multiprocessor jobs receives growing attention in the scheduling
theory [8]. Many computer systems oer some kinds of parallelism. The energy
ecient scheduling of parallel jobs arises in testing and reliable computing, par-
allel applications on graphics cards, computer control systems and others.
2 Related Research
For the preemptive single-processor setting, Yao et al. [14] developed a polyno-
mial time algorithm, that outputs a minimum energy schedule. The preemptive
multiprocessor scheduling of single-processor jobs has been widely studied, see
e.g. [1,4,6,13]. The authors proposed exact polynomial algorithms for this prob-
lem with migration. The works [1,4,13] are based on dierent reductions of the
problem to maximum ow problems. As far as we know, the algorithm presented
in [13] has the best running time among the above-mentioned algorithms.
Albers et al. [3] studied the preemptive problem on parallel processors where
the migration of jobs among processors is disallowed. They showed that if all jobs
have unit work and deadlines are agreeable, an optimal schedule can be com-
puted in polynomial time. At the same time, the general speed scaling scheduling
problem with unit-work jobs was proved to be NP-hard, even on two proces-
sors. A common rule to design algorithms for problems without migration is to
rst dene some strategy that assigns jobs to processors, and then schedule the
assigned jobs separately
on each processor. Albers et al. [3] presented using this
rule an 24 -approximation algorithm for instances with agreeable deadlines,

1
and an 2 2 m -approximation algorithm for instances with common release
dates, or common deadlines. Greiner et al. [10] showed that any -approximation
algorithm for parallel processors with migration can be transformed into a B -
approximation algorithm for parallel processors without migration, where B
is the -th Bell number. The result holds when m.
Bampis et al. [5] considered the problem on heterogeneous processors with
preemption. They assume that each processor i has its own power function,
s(i) , and jobs characteristics are processor dependent. For the case where job
migrations are allowed, an algorithm has been proposed, that returns a solution
within an additive error in time polynomial in the problem size and in 1 . They
also developed an approximation algorithm of ratio (1 + ) B for the problem
without migration, where B is the generalized Bell number [5]. Recently, Albers
et al. [2] proposed a faster combinatorial algorithm based on ows for preemptive
scheduling of jobs whose density is lower bounded by a small constant, and the
migration is allowed.
An Approximation Algorithm for Preemptive Speed Scaling Scheduling 353
We extend the study of speed scaling scheduling to the case of parallel

jobs. Recently, it has been proved that the speed scaling scheduling problem
of rigid jobs is NP-hard even both the preemption and the migration of jobs are
allowed [12]. An approximation algorithm has been proposed in [12] that returns
a solution within an additive error > 0 and runs in time polynomial in m, 1/
and the input size. Note that the algorithm is pseudopolynomial and it is based
on solving a conguration linear program using the Ellipsoid method which is
rather complicated and not ecient in practice. In the current paper we propose

1 1
strongly polynomial 2 m -approximation algorithm for scheduling of rigid
jobs when the preemption and the job migrations are allowed.
3 Our Result
Here we consider the speed scaling scheduling problem of rigid jobs with migra-

1 1
tion and present 2 m -approximation algorithm for this problem.
Our algorithm consists of two stages. At the rst stage we solve an auxiliary
min-cost max-ow problem in order to obtain a lower bound on the minimal
energy consumption and an assignment of the jobs to time intervals. At this
stage we follow the approach proposed in [13]. Then, at the second stage, we
determine speeds of jobs and schedule them separately for each time interval.
The first stage. Due to the convexity of the speed-to-power function, the energy
consumption is minimized if each job j is processed with a xed speed sj , which
does not change during the processing of the job. Therefore, we can formulate
the problem with the variables pj = Wj /sj , where pj is treated as an actual
processing time of job j J . The objective function is written as follows:
n
Wj
F = pj sizej .
j=1
pj
Let us divide the interval [minj rj , maxj dj ] into subintervals Ik = [tk1 , tk ] by

using the release dates rj and the deadlines dj for j J as break-points. Denote
the length of interval Ik by k , k = 1, . . . , , where 2n1. For a job j, denote
the set of the available intervals by (j), where (j) = {Ik : Ik [rj , dj ]}.
Let us construct a bipartite network G = (V, A). The set of nodes is given by
V = {s, t} J I, where J is the set of job nodes and I = {I1 , . . . , I } is the
set of interval nodes. The set of arcs A is given as A = As A0 At , where As =
{(s, j) : j J }, A0 = {(j, Ik ) : j J , Ik (j)}, At = {(Ik , t) : Ik I}, so
that the source s is connected to each job node, each interval node is connected
to the sink t, and each job node j is connected to the nodes Ik associated with
the available intervals (j). We dene the arc capacities as follows:
(s, j) = +, (s, j) As ,
(j, Ik ) = sizej
k , (j, Ik ) A0 , (1)
(Ik , t) = m
k , (Ik , t) At . (2)
x(s,j)
We denote by x(u, v) the amount of ow on an arc (u, v). Note that pj = sizej
x(j,Ik )
denes a total duration of job j and pj,Ik = species a processing time of
sizej

Wj sizej
job j in the interval Ik . Let the cost of ow x(s, j) is x(s, j) x(s,j) , which
is a convex function with respect to x(s, j). The cost of ow on all other arcs is
set to be zero. Then the considered problem reduces to nding a maximum s t
ow in G = (V, A), that minimizes the total cost
n

Wj sizej
x(s, j) .
j=1
x(s, j)
As shown in [13] the obtained min-cost max-ow problem can be solved in

O(n3 ) time. It is not dicult to prove that any feasible schedule species a feasi-
ble ow with the same cost in the network G. The converse is not true, as we shall
see in Example 1, below. At the second stage we use the preemptive sizej -list-
scheduling algorithm [11] to construct a feasible schedule in each interval I I.
The second stage. Let I be an arbitrary time interval and its length is equal
to . We denote by J the subset of jobs, which are assigned to time interval I
at the rst stage, i.e. J = {j J : x(j, I) > 0}. The capacity constraints (1)
and (2) imply that pj,I = x(j, I)/sizej and jJ pj,I sizej m.
The preemptive sizej -list-scheduling algorithm for an interval I works as
follows. At every decision point (i.e., the start time of the interval or completion
time of a job) all currently running jobs are interrupted. Then not yet completed
jobs are considered in order of nonincreasing sizej -values, and as many of them
are greedily assigned to the processors as feasibly possible. The time complexity
of the algorithm is O(n2 ).
We claim that the length of the constructed schedule is at most 2
1
m
(see Lemma 1 below). By increasing the speed of each job in 2 m 1
times
we obtain a schedule of the length at most . The total energy consumption

1 1
for interval I is increased by a factor 2 m . The nal schedule is con-
structed by combining the schedules found for each individual interval, so we

1 1
get 2 m -approximate solution of the original speed scaling problem with
rigid jobs. We note that the number of intervals does not exceed 2n 1 and,
hence, the running time of the second stage is O(n3 ). As a result, we have

1 1
Theorem 1. A 2 m -approximate schedule can be found in O(n3 ) time
for the preemptive speed scaling problem of rigid jobs with migration.
Now we prove Lemma 1.
Lemma 1. Given m processors, an interval I of duration , and a set of jobs

J with processing times pj,I and sizes sizej , where jJ pj,I sizej m.
The length of the schedule
constructed
by the preemptive sizej -list-scheduling
algorithm is at most 2 m 1
.
Proof. Let l be the last job in the preemptive list-schedule (if there are several
such jobs, we choose a job with the smallest value sizej ), and let Cl be its com-
pletion time (the length of the schedule). We consider two cases: (I) sizel > m 2
and (II) sizel m 2.
Case (I): sizel > m 2 . According to the preemptive sizej -list-scheduling
algorithm we obtain that exactly one job is executed at each time moment and
m2
sizej sizel m+1 = (m+0.5)(m0.5) + 14 (m+0.5)(m0.5)+0.25 = 2m1 for
2 2(m0.5)
m
2(m0.5)
all j J . It follows that jJ pj,I sizej 21/m p j,I . From (2) we get
m
jJ
21/m jJ p j,I m and jJ p j,I 2 1
m .
Case (II): sizel m 2 . We claim that, at every point in time during the sched-
ule, either job l is undergoing processing on some sizel processors or job l is not
executed and at least (msizel +1) processors are busy. Therefore, the total load
of all processors jJ pj,I sizej is at least pl,I sizel + (Cl pl,I )(m sizel + 1).

Suppose that Cl > 2 m 1
, then we get the inequality

1
pj,I sizej > pl,I sizel + 2 pl,I (m sizel + 1)

m
jJ

= m + ( pl,I ) (m 2sizel + 1) + (sizel 1) m,
m
which leads to a contradiction.

As shown in [11], the approximation ratio of 2 m 1
for the preemptive
sizej -list-scheduling algorithm is tight even if sizej = 1 for all jobs. As a result,

1 1
the energy consumption is increased in 2 m times when we put the result-
ing schedule inside the interval I. Now we show that the approximation ratio of
our algorithm can not be improved even if we will use an exact algorithm for
minimization of makespan at the second stage.
Example 1. Given m big jobs j = 1, . . . , m of work Wj = m 1 and size sizej =

m and m small jobs j = m + 1, . . . , 2m with work Wj = m and size sizej = 1.
The release dates of all jobs are equal to 0. All small jobs have the common
deadline m2 and the deadline of big job j is dj = mj, j = 1, . . . , m.
Optimal solution has energy consumption m3 and length m2 , by scheduling
the big jobs from time 0 to time m(m 1) and using the last interval [m(m
1), m2 ) for the small jobs. The speed of each processor is equal to 1.
Consider the following optimal solution of the min-cost max-ow problem.
Let exactly one big job j and one small job m + j be assigned to interval [m(j
1); mj), j = 1, . . . , m. Though the total load of each pair jobs j and m + j is m2 ,
it is required 2m 1 time units to execute these jobs with speed 1. Thus, after
increasing the speed of each processor at each time point in 2 m 1
times, the

1 1
total energy consumption is increased by a factor 2 m .
4 Conclusion
We study the energy minimization problem of scheduling rigid jobs on m speed
scalable processors. For migratory case of the problem we propose a strongly
polynomial time approximation algorithm based on a reduction to the min-cost

1 1
max-ow problem. The algorithm has approximation ratio 2 m and this
bound is tight. Our result can be generalized to the case of job-dependent energy
consumption when each job j has its own constant j > 1. For this case our

1 1
algorithm obtains the 2 m -approximate solution where = max j .
jJ
Acknowledgements. This research is supported by the Russian Science Foundation

grant 15-11-10009.
References
1. Albers, S., Antoniadis, A., Greiner, G.: On multi-processor speed scaling with
migration. J. Comput. Syst. Sci. 81, 11941209 (2015)
2. Albers, S., Bampis, E., Letsios, D., Lucarelli, G., Stotz, R.: Scheduling on power-
heterogeneous processors. In: Kranakis, E., Navarro, G., Chavez, E. (eds.) LATIN
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3. Albers, S., Muller, F., Schmelzer, S.: Speed scaling on parallel processors. In: 19th
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Learning and Intelligent Optimization
for Material Design Innovation
Amir Mosavi1,2(&) and Timon Rabczuk1(&)

1
Institute of Structural Mechanics, Bauhaus-Universitat Weimar,
Marienstr.15, 99423 Weimar, Germany
{amir.mosavi,timon.rabczuk}@uni-weimar.de
2
Department of Computer and Information Science,
Norwegian University of Science and Technology,
Sem Saelandsvei 9, 7491 Trondheim, Norway
Abstract. Learning and intelligent optimization (LION) techniques enable

problem-specic solvers with vast potential applications in industry and busi-
ness. This paper explores such potentials for material design innovation and
presents a review of the state of the art and a proposal of a method to use LION
in this context. The research on material design innovation is crucial for the
long-lasting success of any technological sector and industry and it is a rapidly
evolving eld of challenges and opportunities aiming at development and
application of multi-scale methods to simulate, predict and select innovative
materials with high accuracy. The LION way is proposed as an adaptive solver
toolbox for the virtual optimal design and simulation of innovative materials to
model the fundamental properties and behavior of a wide range of multi-scale
materials design problems.
Keywords: Machine learning Optimization Material design
1 Introduction
Materials design is crucial for the long-lasting success of any technological sector, and
yet every technology is founded upon a particular materials design set. This is why the
pressure on development of new high-performance materials for use as high-tech
structural and functional components has become greater than ever. Although the
demand for materials is endlessly growing, experimental materials design is attached to
high costs and time-consuming procedures of synthesis. Consequently simulation
technologies have become completely essential for material design innovation [1].
Naturally the research community highly supports the advancement of simulation
technologies as it represents a massive platform for further development of scientic
methods and techniques. Yet computational material design innovation is a new
paradigm in which the usual route of materials selection is enhanced by concurrent
materials design simulations and computational applications [19].
Designing new materials is a multi-dimensional problem where multiple criteria of
design need to be satised. Consequently material design innovation would require
advanced multiobjective optimization (MOO) [13] and decision-support tools [12].
In addition the performance and behavior of new materials must be predicted in
https://doi.org/10.1007/978-3-319-69404-7_31
Learning and Intelligent Optimization for Material Design Innovation 359
different design scenarios and conditions [2]. In fact predictive analytics and MOO
algorithms are the essential computation tools to tailor the atomic-scale structures,
chemical compositions and microstructures of materials for desired mechanical prop-
erties such as high-strength, high-toughness, high thermal and ionic conductivity, high
irradiation and corrosion resistance [7]. Via manipulating the atomic-scale dislocation,
phase transformation, diffusion, and soft vibrational modes the material behavior on
plasticity, fracture, thermal, and mass transport at the macroscopic level can be pre-
dicted and optimized accurately [17]. Therefore the framework of a predictive
simulation-based optimization of advanced materials, which yet to be realized, repre-
sents a central challenge within material simulation technology [9]. Consequently
material design innovation is facing the ever-growing need to provide a computational
toolbox that allows the development of tailor-made molecules and materials through
the optimization of materials behavior [10]. The goal of such toolbox is to provide
insight over the property of materials associated with their design, synthesis, pro-
cessing, characterization, and utilization [19].
2 Computational Materials Design Innovation
Computational materials design innovation aims at development and application of

multiscale methods to simulate advanced materials with high accuracy [17]. A key to
meet the ever-ongoing demand on increasing performance, quality, specialization, and
price reduction of materials is the availability of simulation tools which are accurate
enough to predict and optimize novel materials on a low computation cost [6]. A major
challenge however would be the hierarchical nature inherent to all materials.
Accordingly to understand a material property on a given length and time scale it is
crucial to optimize and predict the mechanisms on shorter length and time scales all the
way down to the most fundamental mechanisms describing the chemical bond. Con-
sequently the materials systems are to be simultaneously studied under consideration of
underlying nano-structures and Mesomanufacturing Scales. Such design process is
highly nonlinear and requires an interactive MOO toolset [12].
2.1 Interdisciplinary Research and Research Gap

Structure calculations of materials [20], systematic storage of the information in
database repositories [8], materials characterization and selection [18], and gaining new
physical and environmental insights [9] account for big data technologies. In addition
making decision for the optimal materials design needs MOO tools as well as an
efcient decision-support system for post-processing [21]. This is considered as a
design optimization process of the microstructure of materials with respect to desired
properties and Mesoscale functionalities. Such process requires a smart agent which
learns from dataset and makes optimal decisions. The solution of this inverse problem
with the support of the virtual test laboratories and knowledge-based design would be
the foundation of tailor-made molecules and materials toolbox. With such an integrated
toolbox at hand the virtual testing concept and application is realized. This challenging
task can only be accomplished through a variety of scale bridging methods which
360 A. Mosavi and T. Rabczuk
requires machine learning and optimization combined [4]. Furthermore a great deal of
understanding on big data and prediction technologies for microstructure behavior of
existing materials, as well as the ability to test the behavior of new materials at the
atomic, microscopic and mesoscale is desired to condently modifying the materials
properties [7]. Numerical analysis further allows efcient experiments with entirely
new materials and molecules [20]. Basic machine learning technologies such as arti-
cial neural networks [21], and genetic algorithms [9], Bayesian probabilities and
machine learning [8], data mining of spectral decompositions [7], renement and
optimization by cluster expansion [20], structure map analysis and neural networks [1],
and support vector machines [19], have been recently used for this purpose.
Computational materials design innovation to perfect needs to dramatically improve
and put crucial components in place. To be precise, data mining, efcient codes, Big data
technologies, advanced machine learning techniques, intelligent and interactive MOO,
open and distributed networks of repositories, fast and effective descriptors, and
strategies to transfer knowledge to practical implementations are the research gaps to be
addressed [6]. In fact the current solvers rely only on a single algorithm and address
limited scales of the design problems [17]. In addition there is a lack of reliable visu-
alization tools to better involve engineers into the design loop [11]. The absence of
robust design, lack of the post-processing tools for multicriteria decision-making, lack
of Big data tools for an effective consideration of huge materials database are further
research gaps reported in literature [8]. To conclude, the process of computational
material design innovation requires a set of up-to-date solvers to cover a wide range of
problems. Further problem with the current open-source software toolboxes, reported in
[6], is that they require a concrete specication on the mathematical model, and also the
modeling solution is not flexible and adaptive. This has been a reason why the traditional
computation tools for materials design have not been realistic and as effective. Conse-
quently the vision of this work is to propose an interactive toolbox, where the solver
determines the optimal choices via visualization tools as demonstrated in [5]. Ultimately
the purpose is to construct a knowledge-based virtual test laboratory to simultaneously
optimize the hybrid materials microstructure systems, e.g. textile composites. Whether
building atomistic, continuum mechanics or multiscale models, the toolbox can provide
a platform to rearrange the appropriate solver according to the problem at hand. Such
platform contributes in advancement of innovative materials database leading to inno-
vative materials design with the optimal functionality.
3 LION as a Solver
The complex body of information of computational materials design requires the most
recent advancements in machine learning and MOO to scale to the complex and
multiobjective nature of the optimal materials design problems [10]. From this per-
spective the materials design can be seen as a high potential research area and a
continuous source of challenging problems for LION. In the LION way [3] every
individual design task, according to the problem at hand, can be modeled on the basis
of the solvers within the toolbox. To obtain a design model the methodology does not
ask to specify a model, but it experiments with the current system. The appropriate
model is created in the toolbox and further is used to identify a better solution in a
learning cycle. The methodology is based on transferring data to knowledge to optimal
decisions through LION way i.e. a workflow that is referred to as prescriptive analytics
[4]. In addition an efcient Big data application [18] can be integrated to build models
and extract knowledge. Consequently a large database containing the properties of the
existing and hypothetical materials is interrogated in the search of materials with the
desired properties. Knowledge exploits to automate the discovery of improving solu-
tions i.e. connecting insight to decisions and actions [17]. As the result a massively
parallelized multiscale materials modeling tools that expand atomistic-simulation-based
predictive capability is established which leads to rational design of a variety of
innovative materials and applications.
A variety of solvers integrated within the LION include several algorithms for data
mining, machine learning, and predictive analytics which are tuned by cross-validation.
These solvers provide the ability of learning from data, and are empowered by reactive
search optimization (RSO) [4] i.e. the intelligent optimization tool that is integrated into
the solver. The LION way fosters research and development for intelligent optimization
and Reactive Search. Reactive Search stands for the integration of sub-symbolic
machine learning techniques into local search heuristics for solving complex opti-
mization problems via an internal online feedback loop for the self-tuning of critical
parameters [3, 12]. In fact RSO is the effective building block for solving complex
discrete and continuous optimization problems which can cure local minima traps.
Further, cooperating RSO coordinates a collection of interacting solvers which is
adapted in an online manner to the characteristics of the problem. LIONsolver [4],
LIONoso (a non-prot version of LIONsolver), and Grapheur [5], are the software
implementations of the LION way which can be customized for different usage con-
texts in materials design. These implementations have been used for solving a number
of real-life problems including materials selection [18], engineering design [14, 15],
computational mechanics [13], and Robotics [16].
4 Textile Composites Optimal Design
To evaluate the effectiveness of the LION way the case study of textile composites
design with MOO, presented by Milani (2011), is reconsidered using Grapheur. This
case study describes a novel application of LION way dealing with decision conflicts
often seen among design criteria in composites materials design [18]. In this case study
it is necessary to explore optimal design options by simultaneously analyzing materials
properties in a multitude of disciplines, design objectives, and scales. The complexity
increases with considering the fact that the design objective functions are not mathe-
matically available and designer must be in the loop of optimization to evaluate the
Mesomanufacturing Scales of the draping behavior of textile composites. The case
study has a relatively large-scale decision space of electrical, mechanical, weight, cost,
and environmental attributes.
To solve the problem an interactive MOO model is created with Grapheur. With
the aid of the 7D visualization graph the designer in the loop formulates and sys-
tematically compares different alternatives against the large sets of design criteria to
362 A. Mosavi and T. Rabczuk
tackle complex decision-making task of exploring trade-offs and also designing

break-even points. With the designer in the loop, interactive schemes are developed
where Grapheur provides a versatile tool for stochastic local search optimization. Once
the optimal candidates over the ve design objectives preselected, screening the
Mesomanufacturing Scales of draping gures takes place to identify the most suitable
candidate. In this case study the interactive MOO toolset of Grapheur provides a strong
user interface for visualizing the results, facilitating the solution analysis, and
post-processing (Fig. 1).
Fig. 1. 7D visualization graph for MOO and post-processing: Interactive MOO toolset of
Grapheur on exploring trade-offs and simultaneous screening the Mesomanufacturing Scales: the
multi-disciplinary property values of candidate materials are supplied from [12].
5 Conclusions
Computational material design innovation as an emerging area of materials science

requires an adaptive solver to rule a wide range of materials design problems.
The LION way provides a suitable platform for developing a computational toolbox for
the virtual optimal design and simulation-based optimization of advanced materials to
model, simulate, and predict the fundamental properties and behavior of multiscale
materials. The proposed solver is a simple yet powerful concept presenting an inte-
gration of advanced machine learning and intelligent optimization techniques. With a
strong interdisciplinary background the novel application of LION way connects
computer science and engineering, and further strengthens the research direction of
digital engineering.
References
1. Artrith, N.H., Alexander, U.: An implementation of articial neural-network potentials for
atomistic materials simulations. Comput. Mater. Sci. 114, 135150 (2016)
2. Bayer, F.A.: Robust economic Model Predictive Control using stochastic information.
Automatica 74, 151161 (2016)
3. Battiti, R., Brunato, M.: The LION Way: Machine Learning plus Intelligent Optimization.
Lionlab, University of Trento, Italy (2015)
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Proc. VLDB Endow. 6, 11761177 (2013)
5. Brunato, M., Battiti, R.: Grapheur: a software architecture for reactive and interactive
optimization. In: Blum, C., Battiti, R. (eds.) LION 2010. LNCS, vol. 6073, pp. 232246.
Springer, Heidelberg (2010). doi:10.1007/978-3-642-13800-3_26
6. Ceder, G.: Opportunities and challenges for rst-principles materials design and applications
to Li battery materials. Mater. Res. Soc. Bull. 35, 693701 (2010)
7. Fischer, C.: Predicting crystal structure by merging data mining with quantum mechanics.
Nat. Mater. 5, 641646 (2006)
8. Jain, A.: A high-throughput infrastructure for density functional theory calculations.
Comput. Mater. Sci. 50, 22952310 (2011)
9. Johannesson, G.H.: Combined electronic structure and evolutionary search approach to
materials design. Phys. Rev. Lett. 88, 255268 (2002)
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11. Mosavi, A.: Decision-making software architecture; the visualization and data mining
assisted approach. Int. J. Inf. Comput. Sci 3, 1226 (2014)
12. Milani, A.: Multiple criteria decision making with life cycle assessment for material selection
of composites. Express Polym. Lett. 5, 10621074 (2011)
13. Mosavi, A., Vaezipour, A.: Reactive search optimization; application to multiobjective
optimization problems. Appl. Math. 3, 15721582 (2012)
14. Mosavi, A.: A multicriteria decision making environment for engineering design and
production decision-making. Int. J. Comput. Appl. 69, 2638 (2013)
15. Mosavi, A.: Decision-making in complicated geometrical problems. Int. Comput. Appl. 87,
2225 (2014)
16. Mosavi, A., Varkonyi, A.: Learning in Robotics. Int. J. Comput. Appl. 157, 811 (2017)
17. Mosavi, A., Rabczuk, T., Varkonyi-Koczy, A.R.: Reviewing the novel machine learning
tools for materials design. In: Luca, D., Sirghi, L., Costin, C. (eds.) INTER-ACADEMIA
2017: Recent Advances in Technology Research and Education. Advances in Intelligent
Systems and Computing, vol. 660, pp. 5058. Springer, Cham (2018). doi:10.1007/978-3-
319-67459-9_7
18. Mosavi, A., et al.: Multiple criteria decision making integrated with mechanical modeling of
draping for material selection of textile composites. In Proceedings of 15th European
Conference on Composite Materials, Venice, Italy (2012)
19. Saito, T.: Computational Materials Design, vol. 34. Springer Science & Business Media,
Heidelberg (2013)
20. Stucke, D.P., Crespi, V.H.: Predictions of new crystalline states for assemblies of
nanoparticles. Nano Lett. 3, 11831186 (2003)
21. Sumpter, B.G., Noid, D.W.: On the design, analysis, and characterization of materials using
computational neural networks. Annu. Rev. Mater. Sci. 26, 223277 (1996)
Statistical Estimation in Global Random Search
Algorithms in Case of Large Dimensions
Andrey Pepelyshev1,2(B) , Vladimir Kornikov2 , and Anatoly Zhigljavsky1,3

1
School of Mathematics, Cardi University, Cardi CF24 4AG, UK
{pepelyshevan,zhigljavskyaa}@cardiff.ac.uk
2
Faculty of Applied Mathematics, St.Petersburg State University,
Saint Petersburg, Russia
vkornikov@mail.ru
3
Lobachevsky Nizhny Novgorod State University,
23 Prospekt Gagarina, 603950 Nizhny Novgorod, Russia
Abstract. We study asymptotic properties of optimal statistical esti-

mators in global random search algorithms when the dimension of the
feasible domain is large. The results obtained can be helpful in deciding
what sample size is required for achieving a given accuracy of estimation.
Keywords: Global optimization Extreme value Random search

Estimation of end-point
1 Introduction
We consider the problem of global minimization f (x) minxX , where f () is

the objective function and X Rd is a feasible domain. The set X is a compact
set with non-empty interior and the objective function f () is assumed to satisfy
some smoothness conditions which will be discussed below. Let f = minxX f (x)
be the minimal value of f () and x be a global minimizer; that is, x is any
point in X such that f (x ) = f .
If the objective function is given as a black box computer code and there
is no information about this function available of Lipschitz type, then good sto-
chastic approaches often perform better than deterministic algorithms, especially
in large dimensions; see for example [3,4]. Moreover, stochastic algorithms are
usually simpler than deterministic algorithms.
A general Global Random Search (GRS) algorithm constructs a sequence of
random points x1 , x2 , . . . such that the point xj has some probability distribution
Pj , j = 1, 2, . . .; we write this as xj Pj . For each j 2, the distribution Pj
may depend on the previous points x1 , . . . , xj1 and on f (x1 ), . . . , f (xj1 ).
In the present paper, we will mostly concentrate on the so-called Pure Ran-
dom Search (PRS) algorithm, where the points x1 , x2 , . . . are independent and
have the same distribution P = Pj for all j. Simplicity of PRS enables detailed
examination of this algorithm.

https://doi.org/10.1007/978-3-319-69404-7_32
Estimation in Random Search in Large Dimensions 365
2 Statistical Inference About f in Pure Random Search

Consider a PRS with xj P . Statistical inference about f can serve for the
following purposes: (i) devising specic GRS algorithms like the branch and
probability bounds methods, see [2,6] and [4, Sect. 4.3], (ii) constructing stopping
rules, see [5], and (iii) increasing eciency of population-based GRS methods,
see discussion in [3, Sect. 2.6.1]. Moreover, the use of statistical inferences in GRS
algorithms can be very helpful in solving multi-objective optimization problems
with non-convex objectives, see [6].
Since the points xj in PRS are independent identically distributed (i.i.d.)
with distribution P , the elements of the sample Y = {y1 , . . . , yn } with yj =
cumulative distribution function (c.d.f.) F (t) = Pr{x
f (xj ) are i.i.d. with
X : f (x) t} = f (x)t P (dx) = P (W (t f )) , where t f and W () =
{x X : f (x) f + }, 0. Since the analytic form of F (t) is either unknown
or intractable (unless f is very simple), for making statistical inferences about f
we need to use the asymptotic approach based the record values of the sample
Y . It is known that (i) the asymptotic distribution of the order statistics is
unambiguous, (ii) the conditions on F (t) and f () when this asymptotic law
works are very mild and typically hold in real-life problems, (iii) for a broad
class of functions f () and distributions P , the c.d.f. F (t) has the representation
F (t) = c0 (t f ) + o((t f ) ), t f , (1)
where c0 and are some positive constants. The value of c0 is not important but
the value of is essential. The coecient is called tail index and its value is
usually known, as discussed below.
Let be a random variable which has c.d.f. F (t) and y1,n . . . yn,n be
the order statistics for the sample Y . By construction, f is the lower endpoint
of the random variable .
One of the most important result in the theory of extreme order statis-
tics states (see e.g. [3, Sect. 2.3]) that if (1) holds then the c.d.f. F (t) belongs
to the domain of attraction of the Weibull distribution with density (t) =
t1 exp {t } , t > 0 . This distribution has only one parameter, the tail
index .
In PRS we can usually have enough knowledge about f () to get the exact
value of the tail index . Particularly, the following statement holds: if the global
minimizer x of f () is unique and f () is locally quadratic around x then the
representation (1) holds with = d/2. However, if the global minimizer x of
f () is unique and f () is not locally quadratic around x then the representation
(1) may hold with = d. See [4] for a comprehensive description of the related
theory.
The result that has the same order as d when d is large implies the phe-
nomena called the curse of dimensionality. Let us rst illustrate this curse of
dimensionality on a simple numerical example.
366 A. Pepelyshev et al.
3 Numerical Examples
We investigate the minimization problem with the objective function f (x) =
eT1 x, where e1 = (1, 0, . . . , 0)T , and the set X is the unit ball: X = {x
Rd : ||x|| 1}. The minimal value is f = 1 and the global minimizer
z = (1, 0, . . . , 0)T is located at the boundary of X. Consider the PRS algo-
rithm with points xj generated from the uniform distribution PU on X.
Let us give some numerical values. In a simulation with n = 103 and d =
20, we have received y1,n = 0.6435, y2,n = 0.6107, y3,n = 0.6048 and
y4,n = 0.6021. In a simulation with n = 105 and d = 20, we have obtained
y1,n = 0.7437, y2,n = 0.7389, y3,n = 0.7323 and y4,n = 0.726. In Fig. 1 we
depict the dierences yk,n f for k = 1, 4, 10 and n = 103 , . . . , 1013 , where the
horizontal axis has logarithmic scale. We can see that the dierence yk,n y1,n is
much smaller than the dierence y1,n f ; that demonstrates that the problem
of estimating the minimal value of f is very hard.
Fig. 1. Dierences y1,n f (solid), y4,n f (dashed) and y10,n f (dotted), where
yk,n , k = 1, 4, 10, are records of evaluations of the function f (x) = eT1 x at points
x1 , . . . , xn with uniform distribution in the unit hyperball in the dimension d = 20
(left) and d = 50 (right).
Fig. 2. The dierence y1,n f (left) and y10,n y1,n (right) for n = 106 (solid)
and n = 1010 (dashed), where yj,n is the j-th record of evaluations of the function
f (x) = eT1 x at points x1 , . . . , xn with uniform distribution in the unit hyperball in the
dimension d; d varies in [5, 250].
In Fig. 2 we observe that the dierence y1,n f increases as the dimension

d grows, for xed n. Thus, the minimization problem becomes more dicult in
larger dimensions. Also, Fig. 2 shows that dierence y10,n y1,n is much smaller
than the dierence y1,n f .
Consider now the optimal linear estimator based on the use of k order sta-
tistics; this estimator, as shown in [1,4], has the form
1
k
ui
fn,k = yi,n , (2)
Ck, i=1
(i + 2/)
where () is the Gamma-function,

+ 1, i = 1,
ui = ( 1) (i), i = 2, . . . , k 1,

( k 1) (k), i = k,
k
Ck, = i=1 1/i, = 2,
1
2 ( (k + 1)/ (k + 2/) 2/ (1 + 2/)) , = 2.
If the representation (1) holds, then for given k and and as n , the
estimator fn,k is a consistent and asymptotically unbiased estimator of f and
its asymptotic mean squared error E(fn,k f )2 has maximum possible rate
of convergence in the class of all consistent estimators including the maximum
likelihood estimator of f , as shown in [4, Chap. 7]. This mean squared error has
the following asymptotic form:
E(fn,k f )2 = Ck, (c0 n)2/ (1 + o(1)) , n . (3)

2
Using the Taylor series (k + 2/) = (k) + (k) + O(1/2 ) for large
values of , we obtain
1 2((k) 1 + 1/k)
Ck,
+ , (4)
k k
for large , where () = ()/ () is the psi-function. Quality of this approxi-
mation is illustrated on Figs. 3 and 4.
In practice of global optimization, the standard estimator of f is the current
record y1,n = mini=1,...,n f (xi ). Its asymptotic mean squared error is
E(fn,k (e1 ) f )2 = (1 + 2/)(c0 n)2/ (1 + o(1)) , n .
Asymptotic eciency of y1,n is therefore e(y1,n ) = Ck, / (1 + 2/). This

eciency is illustrated on Fig. 5.
GRS algorithms have a very attractive feature in comparison with determin-
istic optimisation procedures. Specically, in GRS algorithms we can use statis-
tical procedures for increasing eciency of the algorithms and devising stopping
368 A. Pepelyshev et al.
Fig. 3. The exact expression of Ck, (solid) and the approximation (4) (dashed) for
k = 2 (left) and k = 10 (right); varies in [5, 50].
Fig. 4. The exact expression of Ck, (solid) and the approximation (4) (dashed) for
= 4 (left) and = 7 (right); as k varies in [2, 25].
Fig. 5. Asymptotic eciency e(y1,n ) of y1,n . Left: k = 2 (solid) and k = 10 (dashed);

as varies in [5, 40]. Right: = 5 (solid) and = 25 (dashed); as k varies in [2, 20].
rules. But do we lose much by choosing the points at random? We claim that if
the dimension d is large then the use of quasi-random points instead of purely
random does not bring any advantage. Let us try to illustrate this using some
simulation experiments.
Using simulation studies we now investigate the performance of the PRS
algorithm with P = PU and quasi-random points generated from the Sobol low-
dispersion sequence.
d We examine the minimization problem with the objective
function f (x) = s=1 (xs | cos(s)|)2 and the set X = [0, 1]d in the dimension
d = 15. In this problem, the global minimum f = 0 is attained at the internal
point x = (| cos(1)|, . . . , | cos(d)|). For each run of the PRS algorithm, we gen-
erate n points and compute the records y1,n and y2,n , for n = 103 , 104 , 105 , 106 .
Fig. 6. Boxplot of records y1,n for 500 runs of the PRS algorithm with points generated
from the Sobol low-dispersion sequence (left) and the uniform distribution (right),
d = 15.
We repeat this procedure 500 times and show the obtained records as boxplots
in Fig. 6.
We can see that the performance of the PRS algorithm with points gener-
ated from the Sobol low-dispersion sequence and the uniform distribution is very
similar. We also note that the variability of y1,n is larger than variability of y4,n
and the dierence y10,n y4,n has a small variability.
Acknowledgements. The work of the rst author was partially supported by the
SPbSU project No. 6.38.435.2015 and the RFFI project No. 17-01-00161. The work
of the third author was supported by the Russian Science Foundation, project No.
15-11-30022 Global optimization, supercomputing computations, and applications.
References
1. Zhigljavsky, A.: Mathematical Theory of Global Random Search. Leningrad Uni-
versity Press (1985). in Russian
2. Zhigljavsky, A.: Branch and probability bound methods for global optimization.
Informatica 1(1), 125140 (1990)
3. Zhigljavsky, A., Zilinskas, A.: Stochastic Global Optimization. Springer, New York
(2008)
4. Zhigljavsky, A.: Theory of Global Random Search. Kluwer Academic Publishers,
Boston (1991)
5. Zhigljavsky, A., Hamilton, E.: Stopping rules in k-adaptive global random search
algorithms. J. Global Optim. 48(1), 8797 (2010)
6. Zilinskas, A., Zhigljavsky, A.: Branch and probability bound methods in
multi-objective optimization. Optim. Lett. 10(2), 341353 (2016). doi:10.1007/
s11590-014-0777-z
A Model of FPGA Massively Parallel
Calculations for Hard Problem of Scheduling
in Transportation Systems
Mikhail Reznikov(&) and Yuri Fedosenko(&)
Volga State University of Water Transport, Nizhny Novgorod, Russia

mirekez@gmail.com, fds@vgavt-nn.ru
Abstract. The FPGA calculation hardware was estimated in terms of perfor-

mance of solving a hard nonlinear discrete optimization problem of scheduling
in transportation systems. The dynamic programming algorithm for a large-scale
problem is considered and a model of its decomposition into a set of smaller
problems is investigated.
Keywords: Discrete optimization Dynamic programming Job-shop

scheduling FPGA calculations Parallel computing
1 Introduction
A discrete nonlinear optimization problem that arises in real-life transportation systems

of river multisectional trains while they pass servicing (loading, unloading or fueling)
terminal (see, e.g., [1]) is considered. This problem is NP-hard and is characterized by an
exponential time complexity depending on the schedule size n (see, e.g., [2, 3]). An
executive planning requires the preparation of a servicing schedule that denes an order
of the vessel processing. Such a schedule can be synthesized by an operator in a strongly
limited time period before a servicing stage begins. From practical considerations, since
one service cycle takes a full day, a one-hour limit was taken for a schedule creating time
in this work. This time period should include dozens of runs of a computational algo-
rithm with slightly different inputs. Since the resources consumption estimation for such
an algorithm is On2n T, where T is the number of discrete time intervals, for even not
high values of n and T as, for example, n = 25 and T = 16, the algorithm would require
more than 13 billion of calculation operations and would take several minutes on a
modern CPU. Thus, it is necessary to search for the best hardware approach and to
investigate computational models providing a signicant performance improvement.
The dynamic programming (DP) approach (see, e.g., [4]) which can be used for
problem solving is memory-limited and its parallelization requires the development of
special algorithms and architectures (see, e.g., [5, 6]). Some of perspective research
directions in the eld of specialized computational systems use multi-core graphical
The reported study was funded by RFBR according to the research project 18-07-01078a.

https://doi.org/10.1007/978-3-319-69404-7_33
A Model of FPGA Massively Parallel Calculations for Hard Problem 371
processing units (GPU), digital signal processors (DSP), neuro-computers, alternative

physical imitation systems (see, e.g., [7]).
A separate direction of the investigation of massively parallel computing engines is
based on the eld programmable gate array (FPGA). The state-of-the-art in the eld of
FPGA-based discrete optimization is presented, for example, in [8, 9] and shows
hardware approach benets for suboptimal solution nding. In comparison with these
works, in the present short paper the way to efciently search for an optimal solution by
using a hardware accelerator is suggested.
2 Service Scheduling Problem
The considered service scheduling problem consists of searching for an optimal

schedule of jobs fz1 ; z2 ; . . .; zn g from a set Z to be processed by a single machine
P. The jobs are characterized by the following parameters i 1; 2; . . .; n:
ti the moment of readiness for processing of the job zi,
si duration of processing of the job zi,
ai(t ti) the linear penalty function (negative weight) for processing started at
time t.
The schedule of processing matches permutation p p1; p2; . . .; pn of
the set of jobs indexes. Let machine P be ready for processing from the time moment
t = 0, processing of each job go without interruptions and simultaneous processing of
several jobs be impossible. Let tk be a moment of processing start for the job with the
index pk k 1; 2; . . .; n. Then the correct schedule will be compact, i.e. t10 tp1
0
and tk0 max tk1 spk1 ; tpk for k = 2, 3, , n.
The problem is to nd the schedule p with the minimal overall penalty over the
whole set Z processed, i.e.
Xn
Wp k1
apk tk0 tpk ! min: 1
As shown, e.g., in [1], problem (1) is strongly NP-hard.
2.1 Dynamic Programming Method

For an efcient problem solution the dynamic programming method is often used
according to the Bellmans principle of optimality (see, e.g., [4]). Let Wkmin t; S be the
minimal overall penalty value for processing a set of jobs S by machine P, which
becomes ready at the moment of discrete time t after the previously processed job p(k).
Let S be a subset of Z with jobs that have been already processed. Then, we can form
the following recurrent expression (counting Wnmin t; Z 0:
0 0
Wkmin t; S min 1 t si ; S [ zi ai t ti ;
Wkmin 2
i 1::n
zi 62 S
372 M. Reznikov and Y. Fedosenko
where t0 maxft; ti g, and the minimal overall penalty is W W0min 0; ;. Thus, the
optimization task can be represented by a recurrent hierarchy of subtasks. The solution
to each task is based on all the solutions to all its subtasks.
The expression (t, S) is called the system state and each expression Wkmin t; S can
be calculated only once to be saved for a later usage.
The algorithm time costs are estimated by counting the overall number of unique
states (t, S). The t value can be limited by some maximum moment T (some dis-
cretization can be chosen for the required accuracy), and the overall number of unique
states will be 2n T. The requirements for the RAM usage will be exponential, too.
3 Parallel Implementation for FPGA
A heterogeneous approach which is considered in this work uses the classical von
Neumann system connected to a special accelerator. Generally, it can be one of several
types like GPU, FPGA or a specialized processor. In this work, we consider the
possibility of using FPGA as a coprocessor to accelerate the optimization task solving.
We suggest a model of massively parallel calculations for a part of the original
problem.
For example, let us consider the original problem with n = 22 and T = 32. The
process of the DP algorithm consists of a continuous calculation of Wkmin t; S values
for all possible system states (t, S) for the stage k from n to 0. Each Wkmin is calculated
on the base of the previously calculated values. The structure of system states realizes
the recurrent hierarchy. As shown on Fig. 1(a), the number of states for processing
depends on its depth in hierarchy Skh marks the set of processed jobs with the serial
number h for the stage k). The CPU calculation time h for each stage is shown on Fig. 1
(b). The most of time is spent for calculation of stages with the number k close to n/2.
Hence, for a parallel realization we can calculate each stage separately and use all the
resources.
Fig. 1. (a) DP states hierarchy by stages; (b) time in seconds needed for each stage solving.
3.1 Decomposition of the Original Problem

Each system state (t, S) corresponds to some subtask with a lower size. The size of the
subtask is equal to n k where k is the size of a set S. Each such subtask can be
solved separately and without any connection with the original task. A local problem
consists of a scheduling of the remaining n k jobs starting from the selected state.
This method is used to allow a larger problem to be solved by using solutions to
smaller problems.
In general, a separate subtask calculation is harder because there is no secondary
usage of the already calculated Wkmin values that can be stored in memory. So, this
method will be efcient only in the case of ability to process much more states per
second than a CPU does. For such a special computing, an FPGA implementation has
been chosen.
The goal of the FPGA architecture is to continuously provide solutions to the small
problem with schedule size m at a frequency of Integrated Circuit (IC), one solution per
each clock cycle. Therefore, the calculation should use sufcient FPGA resources to
process all calculations in parallel. The example of an IC for m = 4 and T = 16 is
shown on Fig. 2(a) for demonstrative purposes. One ALU module is shown in Fig. 2
(b). The IC was built using hardware description language (HDL) by the following
rules:
Fig. 2. (a) ALU DP hierarchy example for m = 4, exact schedules are shown; (b) one ALU.
1. The rst part generates ALU for all possible states (t, S) for t from 0 to T and
S 2 Z.
2. The second part generates busses Wkmin t; S for each ALU and connects the result
registers to all inputs of the other ALUs that require it.
374 M. Reznikov and Y. Fedosenko
This model for the size up to m = 11 has been synthesized for FPGA Virtex-7 565T
(see [10]) with clock 300 MHz, providing one solution to task per clock cycle with
pipelining.
The heterogeneous calculation process works in the following two cycles.
min
1. In the beginning of the solving process the FPGA calculates all Wnm values for the
stage (n-m). Each value is provided as the solution to the subtask of size m.
2. The CPU gets these values from the FPGA, stores in memory tables and then starts
the software-based dynamic programming calculation process from stage (n-m).
As a result of this approach, the CPU skips the rst m stages of the common DP
algorithm calculations. In the case of ability of the FPGA coprocessor to solve a
problem with the size equal to half of the original problem size n, the whole calculation
time can be reduced almost by factor two. Increasing m will lead to the less calculation
time. The problem solution time for a heterogeneous system containing FPGA-based
solver for schedule size m is presented on Fig. 3. The time required for the problem
solution using a 4-core modern CPU is shown as a reference. In particular, the time for
m = 11 was taken from the performance estimation of the heterogeneous system
prototype. The time for m > 11 is the result of mathematical modelling using a sim-
ulator. The original problem size is n = 22 and T = 32.
Fig. 3. Processing time h of the original problem in seconds for different FPGA-based solver
schedule size m compared with a 4-core CPU implementation.
The considered approach allows a linear scaling of DP algorithm for the original
problem solving. However, the maximum size m of the problem which can be solved is
signicantly limited by the available FPGA resources. Taking into account the fact that
the last generation of FPGA contains more resources than devices used in this work, we
can conclude that coprocessors can provide a signicant solving time reduction. By
using several FPGA devices it is possible to provide a linear scalability of the solving
algorithm.
From the other hand, this model demonstrates the way how FPGAs or, generally,
application-specic integrated circuits (ASICs) can be used for a fast solving of
NP-hard discrete optimization problems.
4 Conclusions
As the main result of this work, a special model of a heterogeneous architecture with
the FPGA massively parallel calculations was suggested. The model is based on
decomposition of the original problem using dynamic programming method and pro-
vides the following benets:
a 24 times synthesis time reduction with one FPGA architecture in comparison to
the CPU-based realization,
the linear scalability is possible if more FPGA devices are used.
References
1. Kogan, D.I., Fedosenko, Yu.S.: Optimal servicing strategy design problems for stationary
objects in a one-dimensional working zone of a processor. Autom. Remote Control 71(10),
20582069 (2010)
2. Kogan, D.I., Fedosenko, Yu.S.: The discretization problem: analysis of computational
complexity, and polynomially solvable subclasses. Discrete Math. Appl. 6(5), 435447
(1996)
3. Sammarra, M., Cordeau, J.-F., Laporte, G., Monaco, M.F.: A tabu search heuristic for the
quay crane scheduling problem. J. Sched. 10(4), 327336 (2007)
4. Bellman, R.E., Dreyfus, S.E.: Applied Dynamic Programming. Princeton Univ. Press,
Princeton (1962)
5. Cuencaa, J., Gimnezb, D., Martnez, J.: Heuristics for work distribution of a homogeneous
parallel dynamic programming scheme on heterogeneous systems. Parallel Comput. 31(7),
711735 (2005)
6. Gergel, V.P.: High-Performance Computing for Multi-Processor Multi-Core Systems. MGU
Press, Moscow (2010). (In Russian)
7. Brodtkorb, A.R., Dyken, C., Hagen, T.R., Hjelmervik, J.M., Storaasli, O.O.: State-of-the-art
in Heterogeneous Computing. Scientic Programming 18(1), 133 (2010)
8. Madhavan, A., Sherwood, T., Strukov, D.: Race logic: a hardware acceleration for dynamic
programming algorithms. In: ISCA 2014, Minnesota, USA, pp. 517528 (2014)
9. Tao, F., Zhang, L., Laili, Y.: Job shop scheduling with FPGA-based F4SA. In: Tao, F.,
Zhang, L., Laili, Y. (eds.) Congurable Intelligent Optimization Algorithm. SSAM, pp. 333
347. Springer, Cham (2015). doi:10.1007/978-3-319-08840-2_11
10. 7 Series FPGAs Data Sheet: Overview, March 2017. https://www.xilinx.com/support/
documenta-tion/data_sheets/ds180_7Series_Overview.pdf
Accelerating Gradient Descent with Projective
Response Surface Methodology
Alexander Senov(B)
Faculty of Mathematics and Mechanics, Saint Petersburg State University,

198504 Universitetsky Prospekt, 28, Peterhof, St. Petersburg, Russia
alexander.senov@gmail.com
http://math.spbu.ru/eng
Abstract. We present a new modification of gradient descent algorithm

based on surrogate optimization with projection into low-dimensional
space. It consequently builds an approximation of the target function
in low-dimensional space and takes the approximation optimum point
mapped back to original parameter space as the next parameter estimate.
An additional projection step is used to fight the curse of dimensional-
ity. Major advantage of the proposed modification is that it does not
change gradient descent iterations, thus it may be used with almost any
zero- or first-order iterative method. We give a theoretical motivation
for the proposed algorithm and experimentally illustrate its properties
on modelled data.
Keywords: Least-squares Steepest descent Quadratic program-

ming Projective methods
1 Introduction
Mathematical optimization is a very popular and widely used tool in todays

control, machine learning and other applications since almost any process or
procedure is essentially about maximising or minimising some quantity (see, for
example [5]). In the case of high function computation complexity, unknown
Hessian or high dimensional parameter space, one can choose among many zero-
or rst-order iterative optimization algorithms (see [2,68]), although some of
them share one but yet important drawback: at each iteration they explicitly
consider only the current point and function value estimates ignoring preceding
history and, thus, loosing potentially highly important information.
In contrast, surrogate optimization methods use history of the past points
for approximating the objective function by another function called surrogate
and takes the next estimate based on it (see, e.g., [4]). For example, quadratic
response surface methodology constructs surrogate as a second order polynomial
constructed via polynomial regression and take its minimum as the next esti-
mate (see, e.g., [1]). Despite many advantages, many surrogate models share

https://doi.org/10.1007/978-3-319-69404-7_34
Accelerating Gradient Descent with Projective Approximation 377
two drawbacks: they are memory and time consuming and their performance
depends on the chosen surrogate adequacy.
Additionally, the so-called multi-step optimization methods do use history
too (see, e.g., [8]). Many of them do use only xed amount of history, e.g. two-
step Heavy-ball method use only two past steps. Others do use parametrized
amount of history, like multi-step quasi-Newton method [3], but in slightly dif-
ferent way (e.g., not using the projection trick, or without explicit quadratic
approximation).
In this paper we propose a new method which is essentially an incorporation
of quadratic response surface methodology into the gradient descent algorithm.
To neutralize memory footprint of quadratic polynomial we use a projection
trick. The general idea of the proposed algorithm is to use a sequence of points
obtained from the gradient descent iterations as follows:
1. K1 consecutive points used to train incremental principal component analysis
algorithm which produces orthogonal projection matrix P.
2. K2 consecutive points used to collect training set in low-dimensional space
obtained with P.
3. Quadratic polynomial tted to collected training set and the argument of the
polynomial minimum returned back to the original space used as the next
point estimate.
These steps are executed iteratively producing an additional point every K1 +
K2 gradient descent iterations. We consider gradient descent algorithm as an
example, but it may be easily replaced with some other zero-order or rst-order
iterative optimization method.
The paper is organized as follows. In Sect. 2 we describe the proposed
algorithm which improves gradient descent by using the projective quadratic
response surface methodology. In Sect. 3 we provide theoretical motivation
behind it. Further, in Sect. 4 we report a case study on modelled data and discuss
its results. Finally, Sect. 5 concludes the paper.
2 Algorithm Description
A pseudo-code of the proposed algorithm (Algorithm 1) is given in Fig. 1. It has
the following parameters: f : Rd R function to be optimized; x f : Rd R
function gradient or its approximation (in case of zero-order method); R
step size; d N+ original space dimensionality; q N+ projective space
dimensionality; T N+ number of iterations; IncrPCA incremental PCA
algorithm; K1 N+ number of points for incremental PCA tting; K2 N+
number of points for surrogate construction.
We use x to denote sample mean vector. One might be confused by back-
ward projection step of transforming z in low-dimensional space to x in high-
dimensional space (line 18 in Algorithm 1). This transformation is motivated by
Proposition 1 (Sect. 3). As for the choice of principal component analysis as a
tool for the orthogonal projection construction it is rather motivated by practice
378 A. Senov
Fig. 1. Proposed gradient descent modification
and intuition: the rst q principal components have the largest possible variance.
The more information (in terms of variance) we keep from the original points
the more accurate our approximation will be.
As one can see, the proposed modication utilizes: O (K1 qd) operations and
O (qd) memory
at step (1), O (K2 qd) operations
O (K2 q) memory at step (2)
and
and O K2 q 2 + q 3 operations and O K2 q+ q 2 memory at step (3). Hence,
2modication adds at most O qd + q in the number of operations and
3
this
O q + qd + K2 q in the memory consumption per single gradient descent iter-
ation.
3 Theoretical Background
In this section we provide theoretical motivation behind the proposed algorithm.
Proofs sketches are given in Appendix A.
Proposition 1. Consider P Rqd , PP = Iq and {x1 , . . . , xK } Rd . Then

K
K
1
=
x argmin xt x22 = I P P xt + P
z.
{xRd : Px=
z} t=1 K 1
This proposition motivates the backward projection step in the proposed

algorithm: from set {x Rd : Px = z} we pick the point closest to K previous
gradient descent estimates. Further, the following propositions quantify the eect
of the projection on the function optimum point.
Proposition 2. Consider function f (x) = x Ax + b x + c, where A Rdd ,

A 0, b Rd , c R, P Rqd , PP = Iq , z = Px, v = (I P P)x. Then
1
argmin f (P z + v) = PA1 b.
z 2
Proposition 3. Consider function f (x) = x Ax + b x + c, where A Rdd ,
b Rd , c R and A 0, orthogonal projection matrix P Rqd , q < d,
sequence of gradient descent estimates {xt }K d K
1 R , their projections {zt }1
q K
R , zt = Pxt and corresponding function values {yt }1 , yt = f (xt ).
= 12 P PA1 b + (I P P)x. Then
Let x
2
1 1
argmin f 22
x
= (I P P) A bx
2 .
2
This proposition provides an estimate of the dierence between the true

function optimum point and its estimate obtained after each iteration of the
proposed algorithm (lines 319) and implies the following two important facts.
1 The dierence between the optimum point and obtained estimate lies in the
kernel of orthogonal projection P. Hence, estimate P x
is the best estimate in
terms of this projection.
2 The closer the averaged gradient descent estimates to the optimum points,
benets from precision of gradient descent
the smaller is the error. Hence, x
estimates.
4 A Case Study
First, we describe the modelling strategy. For modelling purposes we use the
following default values: f (x) = x Id x 1Td x, where d is a variable parameter;
d
d 10, T 50, x0 U [0, 1] , = 105 ; q 1, K1 10, K2 5. We vary
parameters T , d, K1 and K2 independently (with other parameters xed) in the
following ranges: T [25, 30, 50, 100]; d [5, 10, 20, 50, 100]; K1 [2, 5, 10, 20];
K2 [2, 5, 10, 20]. For each parameters values combination we execute the gradi-
ent descent algorithm and the proposed algorithm with the same initial estimate
x0 for 103 times and calculate their errors as Euclidean distance from the real
optimum point to the algorithm estimate. Then we calculate a ratio of times
when the proposed algorithm error was less than the gradient descent error.
Table 1 contains the results of the experiment described above. monoton-
ically decreases with T increased since the proposed algorithm works well at
the start but fails to build a surrogate when the gradient descent oscillates near
the optimum point. Further, there are no evident dependency on q since for the
particular function the gradient descent estimates lie on the straight line. Thus,
they are perfectly described by a single dimension. Situation with parameter K1
380 A. Senov
Table 1. Results of the performed numerical experiments. is the ratio of times when
the proposed algorithm error was less than the gradient descent error
Parameter Parameter value

T 25 0.91
T 30 0.90
T 50 0.80
T 100 0.51
d 5 0.81
d 10 0.78
d 20 0.80
d 50 0.81
d 100 0.79
K1 2 0.79
K1 5 0.78
K1 10 0.78
K1 20 0.83
K2 2 0.10
K2 3 0.80
K2 5 0.82
K2 10 0.79
is similar to the parameter q and may be explained by the same reason. Finally,
a huge dierence in quality between K2 = 2 and K2 = 3 is explained by the fact
that one needs at least three points in one-dimensional space to construct the
second order polynomial.
5 Conclusion
We propose a modication to the gradient descent based on the quadratic
response surface methodology with the projection trick. We show that the pro-
posed modication can provide the best optimum approximation with respect
to the considered projection. The synthetic case study shows that the modied
gradient descent can be superior with respect to the original one in terms of the
optimum point estimation error. This modication may be used with other zero-
or rst-order iterative optimization method thus improving their performance.
Acknowledgments. This work was supported by Russian Science Foundation

(project 16-19-00057).
A Proofs
Proof. (of Proposition 1)
K

x argmin xt x22
{xRd : Px=
z} t=1
K

= x || P Pxt + I P P xt P
z + I P P x ||22
t=1
K

= 2(I P P) xt 2K I P P x = 0.
t=1
Since (I P P) is not invertible, the above equation has an innite number of

K
solutions. Hence, we are free to choose any one of them, e.g. x = K
1
1 xt .
Proof. (of Proposition 2)

f (x) = P z + v A P z + v + b P z + v + c

= z PAP z + b + v A P z + v Av b v + c .
Substituting v back and taking derivative with respect to z, we obtain:

0 = 2PAP z + b + x I P P A P
1 1

z = PAP b + x I P P A P
2
1 1 1
= P A b + I P P x = PA1 b.
2 2

Proof. (of Proposition 3) From Propositions 1 and 2: x = I P P x
1 1
2 P PA b. Hence
2
1 1
argmin f 22
x
= A bx
2
2
2
1 1 1
= I P P x + P PA b A b
1

2 2 2
2
1 1
=
(I P P) 2 A b x .
2
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8. Polyak, B.T.: Introduction to Optimization. Translations Series in Mathematics and
Engineering. Optimization Software (1987)
Emmental-Type GKLS-Based Multiextremal
Smooth Test Problems
with Non-linear Constraints
Ya.D. Sergeyev1,2 , D.E. Kvasov1,2 , and M.S. Mukhametzhanov1,2(B)

1
University of Calabria, Rende (CS), Italy
{yaro,kvadim,m.mukhametzhanov}@dimes.unical.it
2
Lobachevsky State University, Nizhni Novgorod, Russia
Abstract. In this paper, multidimensional test problems for methods

solving constrained Lipschitz global optimization problems are proposed.
A new class of GKLS-based multidimensional test problems with con-
tinuously dierentiable multiextremal objective functions and non-linear
constraints is described. In these constrained problems, the global min-
imizer does not coincide with the global minimizer of the respective
unconstrained test problem, and is always located on the boundaries
of the admissible region. Two types of constraints are introduced. The
possibility to choose the diculty of the admissible region is available.
Keywords: GKLS classes of test problems Constrained optimization

Lipschitz global optimization Nonlinear constraints
1 Introduction
Global optimization problems with and without constraints attract a great atten-
tion of researchers from both theoretical and practical viewpoints (see, e.g.,
[3,17,20], for the derivative-free global optimization, [5,9,13,18], for the real-life
engineering problems, [2,17], for the parallel global optimization, etc.).
Let us consider the following constrained problem
f = f (x ) = minf (x), D = {(x1 , ..., xN ) : ai xi bi , i = 1, ..., N } RN ,
xD
(1)
with p constraints
gj (x) 0, j = 1, ..., p, (2)
where the functions f (x) and gj (x), j = 1, ..., p, satisfy the Lipschitz condition
over the hyperinterval D
|f (x ) f (x )| L||x x ||,
(3)
|gj (x ) gj (x )| Lj ||x x ||, j = 1, ..., p, x , x D,

where L and Lj , 0 < L < and 0 < Lj < , j = 1, ..., p, are the Lipschitz
constants for the functions f (x) and gj (x) over the hyperinterval D, respectively
(hereafter || || denotes the Euclidean norm).
https://doi.org/10.1007/978-3-319-69404-7_35
384 Y.D. Sergeyev et al.
There exists a huge number of methods for solving (1)(3) (see, e.g.,
[11,16,21,22]). These methods often have a completely dierent nature and their
numerical comparison can be very dicult (see, e.g., [14,15] for a numerical com-
parison of metaheuristic and deterministic unconstrained global optimization
algorithms). In [19], a new tool called Operational zones for an ecient numeri-
cal comparison of constrained and unconstrained global optimization algorithms
of dierent nature has been proposed. To use it, classes of test problems are
required. On the one hand, there exist many generators of test problems for
global and local optimization (see, e.g., [23] for the landscape generators, [12] for
the multidimensional assignment problem generator, [4,7] for the wide analysis
of dierent test classes and generators, [1,10,16] for dierent classes and gen-
erators of unconstrained test problems). On the other hand, collections of test
problems are used usually in the framework of continuous constrained global
optimization (see, e.g., [6]) due to absence of test classes and generators for such
a type of problems. This paper introduces a new class of test problems with
non-linear constraints, known minimizers, and parameterizable diculty, where
both the objective function and constraints are continuously dierentiable.
2 GKLS Classes of Unconstrained Test Problems
Let us consider the unconstrained GKLS class of test problems with continuously
dierentiable objective function proposed in [8]. Test functions in this class are
generated by dening a convex quadratic function systematically distorted by
cubic polynomials in order to introduce local minima. The objective function
f (x) of the GKLS class is constructed by modifying a paraboloid Z
Z : z(x) = ||x T ||2 + t, x D, (4)
with the minimum value t at the point T int(D), where int(D) denotes the
interior of D, in such a way that the resulting function f (x) has m, m 2,
local minimizers: point T from (4) and points Mi int(D), Mi = T, Mi =
Mj , i, j = 2, ..., m, i = j. The paraboloid Z is modied by cubic polynomials
Ci (x) within balls Si D (not necessarily entirely contained in D) around each
point Mi , i = 2, ..., m (with M1 = T being the vertex of the paraboloid and M2
being the global minimizer of the problem), where
Si = {x RN : ||x Mi || i , i > 0}. (5)
Each class contains 100 test problems and is dened by the following parame-
ters: problem dimension, number of local minima, value of the global minimum,
radius of the attraction region of the global minimizer, and distance from the
global minimizer to the vertex of the paraboloid. An example of the test problem
with 30 local minima is presented in Fig. 1a.
Emmental-Type GKLS-Based Multiextremal Smooth Test Problems 385
(a) (b)
Fig. 1. Original GKLS test function (a) and Emmental-type GKLS-based test function
(b) do not coincide even without constraints.
3 Constrained Emmental-Type Test Problems

In order to have a powerful tool for testing algorithms for constrained global
optimization it is desirable to get test problems satisfying the following condi-
tions:
The global minimizer of the constrained problem is known, randomly gener-
ated, and is placed on the boundary of the feasible region;
The global minimizer of the constrained problem diers from the global min-
imizer of the unconstrained problem.
The constraints are non-linear and satisfy the Lipschitz condition over the
hyperinterval D.
The diculty of the admissible region can be changed.
In order to satisfy the requirements given above, the Emmental-type GKLS-
based class of test problems with smooth objective function is constructed by
the following modications of the GKLS class of unconstrained test problems.
Two types of constraints are proposed, where each constraint gi (x) is the
exterior of a ball with a center Gi and a radius ri , i.e.,
gi (x) = ri ||x Gi || 0, i = 1, ..., p. (6)
The rst-type constraints are constructed as follows. First, the local mini-
mizer (not the global one) Mimin nearest to the vertex of the paraboloid is taken
as follows
imin = arg min {||Mi M1 || i }, (7)
i=3,...,m
where i is the radius of the ball Si (the index i starts from i = 3 since in original
GKLS classes M2 is the global minimizer). Then, the polynomial Cimin around
Mimin is modied in a way, that its minimum value has been set to 2 |f |,
where f is the optimum value of the original unconstrained GKLS problem. The
global minimizer of the Emmental-type unconstrained test problem diers from
the global minimizer of the original GKLS test problem due to this modication
(see Fig. 1b). The ball with the radius r1 = ||M1 Mimin ||imin , and the center
at the vertex of the paraboloid, i.e., G1 = M1 , is taken as the rst constraint.
Second, in order to guarantee that the global minimizer of the constrained
test problem does not coincide with the global minimizer of the unconstrained
modied problem, the ball with the radius r2 = imin and the center G2 = Mimin
is taken as the second constraint.
Then, in order to guarantee that the global minimizer M2 of the constrained
test problem is placed on the boundary, the ball with the radius r3 = 12 2 and
the center at the point G3 = 2||M1rM2 || M1 + (1 2||M1rM2 || )M2 is taken as
the third constraint (see Fig. 2a).
If p1 > 3, where p1 is the number of the rst-type constraints, p1 p, then
the fourth constraint is constructed symmetrically to the third one with respect
to the global minimizer M2 , i.e., r4 = r3 and G4 = M2 + (M2 G3 ) (see Fig. 2b).
The last p1 4 rst-type constraints are taken as p1 4 dierent random
balls Sj , j {3, ..., m}, j = imin, i.e., ri = j(i) and Gi = Mj(i) , i = 5, ..., p1 .
The p2 = p p1 constraints of the second type, where 0 p2 2N , are built
as follows. Random vertices cj = (cj1 , ..., cjN ), cji {ai , bi }, j = 1, ..., p2 , of D are
taken. Then, for each taken vertex cj , the nearest local or global minimum Mi(j)
is found. The (p1 + j)-th constraint is built as a ball with the center Gp1 +j = cj
and the radius rp1 +j = ||cj Mi(j) ||, j = 1, ..., p2 (see Fig. 2c).
The presented Emmental-GKLS class of test problems consists of 100 smooth
objective functions with the same characteristics as the original GKLS-based
test functions, with the global minimum located in a random point with a ran-
dom radius of its region of attraction. Moreover, it is built using p constraints,
including p1 constraints of the rst type, i.e., the constraints related to the local
minima, with 3 p1 m, and p2 = p p1 constraints of the second type, i.e.,
the constraints related to the vertices of D, with 0 p2 2N . The admissible
regions for (p1 , p2 ) = (3, 0), (4, 0), (4, 2), and (20, 2) are presented in Fig. 2.
We can conclude that in the obtained test class:
The global minimizer x of the constrained problem is known, is placed on the
boundaries of the admissible region in a point dierent w.r.t. global minimizer
of the unconstrained Emmental-type GKLS problem.
The global minimizer of the unconstrained Emmental-type GKLS problem
is known. It coincides with the nearest local minimizer to the vertex of the
paraboloid of the original unconstrained GKLS test problem, and diers from
the global minimizer of the constrained test problem.
The diculty of the constrained problem can be easily changed. The simplest
domain has p1 = 3 and p2 = 0 constraints and the hardest one has p1 =
m + 1 and p2 = 2N constraints (notice that the search domain can be simply
connected, biconnected or multiconnected). It should be noticed, that the
global minimizer cannot be an isolated point and is always accessible from a
feasible region having a positive volume.
The constraints are non-linear and satisfy the Lipschitz condition over the
hyperinterval D.
Emmental-Type GKLS-Based Multiextremal Smooth Test Problems 387
Fig. 2. The admissible region and the level curves of the objective function of the
Emmental-type GKLS-based test problem with (a) p1 = 3, p2 = 0; (b) p1 = 4, p2 = 0;
(c) p1 = 4, p2 = 2; (d) p1 = 20, p2 = 2. The admissible region in (d) contains 3 disjoint
subregions. The vertex of the paraboloid is indicated by and the global minimizer of
the constrained problem is indicated by +.
Acknowledgements. This work was supported by the project No. 15-11-30022

Global optimization, supercomputing computations, and applications of the Russian
Science Foundation.
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Author Index
Adriaensen, Steven 3 Kornikov, Vladimir 364

Albu, Alla F. 295 Kotthaus, Helena 180
Archetti, Francesco 34 Kovalenko, Yulia 351
Krylatov, Alexander Yu. 345
Barkalov, Konstantin 18, 301 Kvasov, D.E. 383
Berglund, Erik 307
Bischl, Bernd 180 Lang, Andreas 180
Burke, Edmund K. 93 Lang, Michel 180
Lebedev, Ilya 301
Cceres, Leslie Prez 235 Liberti, Leo 263
Candelieri, Antonio 34 Lpez-Ibez, Manuel 235
Dhaenens, Clarisse 196 Marwedel, Peter 180

Mironov, Sergei V. 251
Erzin, Adil 50 Moons, Filip 3
Evtushenko, Yury G. 295 Mosavi, Amir 358
Mousin, Lucien 196
Fedosenko, Yuri 370 Mukhametzhanov, M.S. 383
Felici, Giovanni 64
Ferone, Daniele 64, 79 Napoletano, Antonio 64, 79
Festa, Paola 64, 79 Now, Ann 3
Nurminski, Evgeni 210
Gergel, Victor 314, 320
Gillard, Jonathan 326 Orlov, Andrei 222
Giordani, Ilaria 34
Goryachih, Alexey 320 Pastore, Tommaso 64
Gretsista, Angeliki 93 Pepelyshev, Andrey 364
Gruzdeva, Tatiana 331
Rabczuk, Timon 358
Hoos, Holger 235 Rahnenfhrer, Jrg 180
Raidl, Gnther R. 150
Kadioglu, Serdar 109 Resende, Mauricio G.C. 79
Karapetyan, Daniel 124 Reznikov, Mikhail 370
Kessaci, Marie-Elonore 196 Richter, Jakob 180
Khamisov, Oleg O. 139 Romito, Francesco 279
Khamisov, Oleg V. 338
Kloimllner, Christian 150 Sellmann, Meinolf 109
Konnov, Igor 166 Senov, Alexander 376
Kononov, Alexander 351 Sergeyev, Ya.D. 383
390 Author Index
Shirokolobova, Anastasiya P. 345 Vernitski, Alexei 124

Sidorov, Sergei P. 251
Stennikov, Valery A. 139
Wagner, Markus 109
Strekalovsky, Alexander 331
Wang, Olivier 263
Strongin, Roman 18
Sttzle, Thomas 235
Zhigljavsky, Anatoly 326, 364
Thomas, Janek 180 Zubov, Vladimir I. 295

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Learning and Intelligent Optimization For Material Design Innovation

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Roberto Battiti

Yaroslav D. Sergeyev (Eds.)

Lobachevsky University of Nizhny Lobachevsky University of Nizhny

Lobachevsky University of Nizhny

ISSN 0302-9743 ISSN 1611-3349 (electronic)

Library of Congress Control Number: 2017956726

LNCS Sublibrary: SL1 Theoretical Computer Science and General Issues

Springer International Publishing AG 2017

Printed on acid-free paper

This Springer imprint is published by Springer Nature

August 2017 Roberto Battiti

Technical Program Committee

Antonio Fuduli University of Calabria, Italy

ric Taillard University of Applied Science of Western Switzerland

Steven Adriaensen Marat Mukhametzhanov

Local Organization Committee

Acknowledgements. The organization of the LION-11 conference was supported by

An Importance Sampling Approach to the Estimation

Test Problems for Parallel Algorithms of Constrained Global Optimization. . . 18

Automatic Configuration of Kernel-Based Clustering:

Solution of the Convergecast Scheduling Problem on a Square

A GRASP for the Minimum Cost SAT Problem . . . . . . . . . . . . . . . . . . . . . 64

An Iterated Local Search Framework with Adaptive Operator

Learning a Reactive Restart Strategy to Improve Stochastic Search . . . . . . . . 109

Efficient Adaptive Implementation of the Serial Schedule Generation

Interior Point and Newton Methods in Solving High Dimensional

Hierarchical Clustering and Multilevel Refinement for the Bike-Sharing

Decomposition Descent Method for Limit Optimization Problems . . . . . . . . . 166

RAMBO: Resource-Aware Model-Based Optimization with Scheduling

A New Constructive Heuristic for the No-Wait Flowshop

Sharp Penalty Mappings for Variational Inequality Problems . . . . . . . . . . . . 210

A Nonconvex Optimization Approach to Quadratic Bilevel Problems . . . . . . 222

An Experimental Study of Adaptive Capping in irace . . . . . . . . . . . . . . . . . 235

Duality Gap Analysis of Weak Relaxed Greedy Algorithms . . . . . . . . . . . . . 251

Controlling Some Statistical Properties of Business Rules Programs . . . . . . . 263

Hybridization and Discretization Techniques to Speed Up Genetic

Identification of Discontinuous Thermal Conductivity Coefficient

Comparing Two Approaches for Solving Constrained Global

Towards a Universal Modeller of Chaotic Systems . . . . . . . . . . . . . . . . . . . 307

An Approach for Generating Test Problems of Constrained

Global Optimization Using Numerical Approximations of Derivatives . . . . . . 320

Global Optimization Challenges in Structured Low Rank Approximation . . . . 326

A D.C. Programming Approach to Fractional Problems . . . . . . . . . . . . . . . . 331

Objective Function Decomposition in Global Optimization . . . . . . . . . . . . . . 338

Projection Approach Versus Gradient Descent for Networks Flows

An Approximation Algorithm for Preemptive Speed Scaling Scheduling

Learning and Intelligent Optimization for Material Design Innovation . . . . . . 358

Statistical Estimation in Global Random Search Algorithms

A Model of FPGA Massively Parallel Calculations

Accelerating Gradient Descent with Projective Response

Emmental-Type GKLS-Based Multiextremal Smooth Test Problems

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Steven Adriaensen(B) , Filip Moons, and Ann Nowe

Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

Abstract. In this paper we consider the problem of estimating the rel-

Keywords: Performance evaluation Algorithms Importance Sam-

requires evaluating each design numerous times, as we are typically interested

Conditional parameters in algorithm conguration [10]: Often not all para-

2 The Algorithm Design Problem (ADP)

|gj (x1 ) gj (x2 )| Lj y1 y2 , 1 j m + 1, y1 , y2 D.