research-article

A Derivative-Free Approach to Constrained Multiobjective Nonsmooth Optimization

Authors:

F. RinaldiAuthors Info & Claims

SIAM Journal on Optimization, Volume 26, Issue 4

Pages 2744 - 2774

https://doi.org/10.1137/15M1037810

Published: 01 January 2016 Publication History

Abstract

In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. We define a linesearch-based solution method, and we show that it converges to a set of Pareto stationary points. To this aim, we carry out a theoretical analysis of the problem by only assuming Lipschitz continuity of the functions; more specifically, we give new optimality conditions that take explicitly into account the bound constraints, and prove that the original problem is equivalent to a bound constrained problem obtained by penalizing the nonlinear constraints with an exact merit function. Finally, we present the results of some numerical experiments on bound constrained and nonlinearly constrained problems, showing that our approach is promising when compared to a state-of-the-art method from the literature.

References

[1]

C. Audet, G. Savard, and W. Zghal, Multiobjective optimization through a series of single-objective formulations, SIAM J. Optim., 19 (2008), pp. 188--210.

[2]

S. Bandyopadhyay, S. K. Pal, and B. Aruna, Multiobjective GAs, quantitative indices, and pattern classification, IEEE Trans. Syst. Man Cybernet. Part B Cybernet. 34 (2004), pp. 2088--2099.

[3]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[4]

A. Conn, K. Scheinberg, and L. N. Vicente, Introduction to Derivative-Free Optimization, MPS-SIAM Ser. Optim. 8, SIAM, Philadelphia, 2009.

[5]

A. L. Custódio, M. Emmerich, and J. F. A. Madeira, Recent developments in derivative-free multiobjective optimization, Comput. Technol. Rev., 5 (2012), pp. 1--30.

[6]

A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente, Errata to Direct Multisearch for Multiobjective Optimization, http://www.mat.uc.pt/~lnv/papers/errata-dms.pdf.

[7]

A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente, Direct multisearch for multiobjective optimization, SIAM J. Optim., 21 (2011), pp. 1109--1140.

[8]

K. Deb, A. Pratap, S. Agarwal, and T. A. M. T. Meyarivan, A fast and elitist multiobjective genetic algorithm: IEEE Trans. Evol. Comput., 6 (2002), pp. 182--197.

[9]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), pp. 201--213.

[10]

G. Fasano, G. Liuzzi, S. Lucidi, and F. Rinaldi, A linesearch-based derivative-free approach for nonsmooth constrained optimization, SIAM J. Optim., 24 (2014), pp. 959--992.

[11]

E. H. Fukuda, L. M. Gran͂a Drummond, and F. M. P. Raupp, An external penalty-type method for multicriteria, TOP, 24 (2016), pp. 493--513.

[12]

T. Glad and E. Polak, A multiplier method with automatic limitation of penalty growth, Math. Program., 17 (1979), pp. 140--155.

[13]

J. B. Hiriart-Urruty, On optimality conditions in nondifferentiable programming, Math. Program., 14 (1978), pp. 73--86.

[14]

X. X. Huang and X. Q. Yang, Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization, SIAM J. Optim., 13 (2002), pp. 675--692.

[15]

Y. Ishizuka and K. Shimizu, Necessary and sufficient conditions for the efficient solutions of nondifferentiable multiobjective problems, IEEE Trans. Syst. Man Cybernet., SMC-14 (1984), pp. 624--629.

[16]

J. Jahn, Introduction to the Theory of Nonlinear Optimization, Springer, Berlin, 1996.

[17]

J. Jahn, Vector Optimization, Springer, Berlin, 2009.

[18]

N. Karmitsa, Test Problems for Large-Scale Nonsmooth Minimization, Technical report No., B. 4/2007, Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland, 2007.

[19]

C.-J. Lin, S. Lucidi, L. Palagi, A. Risi, and M. Sciandrone, Decomposition algorithm model for singly linearly-constrained problems subject to lower and upper bounds, J. Optim. Theory Appl., 141 (2009), pp. 107--126.

[20]

G. Liuzzi, S. Lucidi, F. Parasiliti, and M. Villani, Multiobjective optimization techniques for the design of induction motors, IEEE Trans. Magnet., 39 (2003), pp. 1261--1264.

[21]

S. Lucidi, New results on a continuously differentiable exact penalty function, SIAM J. Optim., 2 (1992), pp. 558--574.

[22]

O. L. Mangasarian, Nonlinear Programming, Classics Appl. Math., SIAM, Philadelphia, 1994.

[23]

K. Shimizu, Y. Ishizuka, and J. F. Bard, Nondifferentiable and Two-level Mathematical Programming, Kluwer, Norwell, MA, 1997.

[24]

A. Suppapitnarm, K. A. Seffen, G. T. Parks, and P. J. Clarkson, A simulated annealing algorithm for multiobjective optimization, Eng. Optim., 33 (2000), pp. 59--85.

[25]

E. Zitzler and L. Thiele, Multiobjective Optimization using Evolutionary Algorithms --- A Comparative Case Study, in Parallel Problem Solving from Nature --- PPSN V: 5th International Conference Amsterdam, The Netherlands, A. E. Eiben, T. Bäck, M. Schoenauer, and H.-P. Schwefel, eds., Springer, Berlin 1998, pp. 292--301.

Cited By

Cristofari ADe Santis MLucidi S(2024)On Necessary Optimality Conditions for Sets of Points in Multiobjective OptimizationJournal of Optimization Theory and Applications10.1007/s10957-024-02478-3203:1(126-145)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10957-024-02478-3
Bigeon JLe Digabel SSalomon L(2024)Handling of constraints in multiobjective blackbox optimizationComputational Optimization and Applications10.1007/s10589-024-00588-289:1(69-113)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1007/s10589-024-00588-2
Chang TWatson LLarson JNeveu NThacker WDeshpande SLux T(2022)Algorithm 1028: VTMOP: Solver for Blackbox Multiobjective Optimization ProblemsACM Transactions on Mathematical Software10.1145/352925848:3(1-34)Online publication date: 10-Sep-2022
https://dl.acm.org/doi/10.1145/3529258
Show More Cited By

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cover image SIAM Journal on Optimization

SIAM Journal on Optimization Volume 26, Issue 4

DOI:10.1137/sjope8.26.4

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© 2016, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2016

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Cited By

Cristofari ADe Santis MLucidi S(2024)On Necessary Optimality Conditions for Sets of Points in Multiobjective OptimizationJournal of Optimization Theory and Applications10.1007/s10957-024-02478-3203:1(126-145)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10957-024-02478-3
Bigeon JLe Digabel SSalomon L(2024)Handling of constraints in multiobjective blackbox optimizationComputational Optimization and Applications10.1007/s10589-024-00588-289:1(69-113)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1007/s10589-024-00588-2
Chang TWatson LLarson JNeveu NThacker WDeshpande SLux T(2022)Algorithm 1028: VTMOP: Solver for Blackbox Multiobjective Optimization ProblemsACM Transactions on Mathematical Software10.1145/352925848:3(1-34)Online publication date: 10-Sep-2022
https://dl.acm.org/doi/10.1145/3529258
Tavares SBrás CCustódio ADuarte VMedeiros P(2022)Parallel strategies for Direct MultisearchNumerical Algorithms10.1007/s11075-022-01364-192:3(1757-1788)Online publication date: 30-Jul-2022
https://dl.acm.org/doi/10.1007/s11075-022-01364-1
Pellegrini RSerani ALiuzzi GRinaldi FLucidi SCampana EIemma UDiez M(2017)Hybrid Global/Local Derivative-Free Multi-objective Optimization via Deterministic Particle Swarm with Local LinesearchMachine Learning, Optimization, and Big Data10.1007/978-3-319-72926-8_17(198-209)Online publication date: 14-Sep-2017
https://dl.acm.org/doi/10.1007/978-3-319-72926-8_17

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