Related papers: Detecting optimality and extracting solutions in p…
We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering.…
We consider polynomial optimization problems on Cartesian products of basic compact semialgebraic sets. The solution of such problems can be approximated as closely as desired by hierarchies of semidefinite programming relaxations, based on…
This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization…
Given polynomials f(x), g_i(x), h_j(x), we study how to minimize f on the semialgebraic set S = { x \in R^n: h_1(x)=...=h_{m_1}(x) =0, g_1(x) >= 0, ..., g_{m_2}(x) >= 0}. Let f_{min} be the minimum of f on S. Suppose S is nonsingular and…
In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best…
Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general,…
Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional…
Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we…
This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing…
We consider the problem of finding the global optimum of a real-valued complex polynomial on a compact set defined by real-valued complex polynomial inequalities. It reduces to solving a sequence of complex semidefinite programming…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is defined also by a PMI. These can be viewed as matrix generalizations of semi-infinite…
In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu [Comptes Rendus de l'Acad\'emie des Sciences-Series I-Mathematics, 328(6) (1999) pp.…
Comparison of Lasserre's measure--based bounds for polynomial optimization to bounds obtained by simulated annealing. We consider the problem of minimizing a continuous function $f$ over a compact set $\mathbf{K}$. We compare the hierarchy…
The Max-k-Cut problem is a fundamental combinatorial optimization challenge that generalizes the classic NP-complete Max-Cut problem. While relaxation techniques are commonly employed to tackle Max-k-Cut, they often lack guarantees of…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…
In this work, the combine the theory of generalized critical values with the theory of iterated rings of bounded elements (real holomorphy rings). We consider the problem of computing the global infimum of a real polynomial in several…
This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the finite convergence of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second…
We propose the first general and practical framework to design certifiable algorithms for robust geometric perception in the presence of a large amount of outliers. We investigate the use of a truncated least squares (TLS) cost function,…
We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof…