Related papers: Polynomial Optimization Over Unions of Sets
Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the…
This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in $\re^m$ defined by finitely many polynomial equations and inequalities, using the Karush-Kuhn-Tucker (KKT) system and sum of squares…
In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
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.…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
SOSOPT is a Matlab toolbox for formulating and solving Sum-of-Squares (SOS) polynomial optimizations. This document briefly describes the use and functionality of this toolbox. Section 1 introduces the problem formulations for SOS tests,…
This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we…
In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as…
This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this…
The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
In this paper, we study the problem of computing by relaxation hierarchies the infimum of a real polynomial function f on a closed basic semialgebraic set and the points where this infimum is reached, if they exist. We show that when the…
We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves…
We consider the unconstrained optimization of multivariate trigonometric polynomials by the sum-of-squares hierarchy of lower bounds. We first show a convergence rate of $O(1/s^2)$ for the relaxation with degree $s$ without any assumption…
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
Given a large data matrix $A\in\mathbb{R}^{n\times n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}\sim P_0$, or instead $A$ contains a principal submatrix $A_{{\sf…
This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors,…
Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set K. The idea consists of approximating from above the indicator function of K with a sequence of…
We consider the quadratic optimization problem $\max_{x \in C}\ x^T Q x + q^T x$, where $C\subseteq\mathbb{R}^n$ is a box and $r := \mathrm{rank}(Q)$ is assumed to be $\mathcal{O}(1)$ (i.e., fixed). We show that this case can be solved in…