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We investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph G=(V1∪V2,E), that is, pairs (A, B) where A⊆V1, B⊆V2, and A∪B are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality |A∪B|, ...

This paper studies Nash equilibrium problems that are given by polynomial functions. We formulate efficient polynomial optimization problems for computing Nash equilibria. The Moment-sum-of-squares relaxations are used to solve them. Under generic ...

The use of machine learning techniques to improve the performance of branch-and-bound optimization algorithms is a very active area in the context of mixed integer linear problems, but little has been done for nonlinear optimization. To bridge this gap, ...

We study separable plus quadratic (SPQ) polynomials, that is, polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials ...

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the ...

De Klerk and Pasechnik introduced in 2002 semidefinite bounds ϑ(r)(G)(r≥0) for the stability number α(G) of a graph G and conjectured their exactness at order r=α(G)−1. These bounds rely on the conic approximations Kn(r) introduced by Parrilo in 2000 for the copositive cone COPn. A ...

In this paper, we establish an equivalence relation between partially symmetric tensors and homogeneous polynomials, prove that every partially symmetric tensor has a partially symmetric canonical polyadic (CP)-decomposition, and present three ...

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-...

Dual decomposition approaches in nonconvex optimization may suffer from a duality gap. This poses a challenge when applying them directly to nonconvex problems such as MAP-inference in a Markov random field with continuous state spaces. To eliminate such ...

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, ...

We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim., 12 (2002), pp. 875--892], ...

In this article, we show that each semidefinite relaxation of a ball-constrained noncommutative polynomial optimization problem can be cast as a semidefinite program with a constant trace matrix variable. We then demonstrate how this constant trace ...

This paper is concerned with minimizing a sum of rational functions over a compact set of high dimension. Our approach relies on the second Lasserre hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward measure in order to ...

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of unions of edges. These polynomials arise naturally to model job-occupancy in some ...

This work is a follow-up and a complement to [J. Wang, V. Magron and J. B. Lasserre, preprint, arXiv:1912.08899, 2019] where the TSSOS hierarchy was proposed for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose ...

This paper is concerned with polynomial optimization problems. We show how to exploit term (or monomial) sparsity of the input polynomials to obtain a new converging hierarchy of semidefinite programming relaxations. The novelty (and distinguishing feature) ...

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a ...

We provide several positive and negative complexity results for solving games with imperfect recall. Using a one-to-one correspondence between these games on one side and multivariate polynomials on the other side, we show that solving games with ...

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre, and a related hierarchy by de Klerk, Hess, and Laurent. For polynomial optimization over the hypercube, we show a refined convergence ...

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on ...