A new look at nonnegativity on closed sets and polynomial optimization

TL;DR

This paper establishes a new characterization of nonnegativity on closed sets using moment matrices and introduces an explicit hierarchy of semidefinite approximations for polynomial nonnegativity, enhancing polynomial optimization methods.

Contribution

It provides a novel, convergent hierarchy of semidefinite relaxations for nonnegative polynomials on closed sets without lifting, improving optimization techniques.

Findings

Characterization of nonnegativity via moment matrices of signed measures.

An explicit hierarchy of semidefinite outer approximations with convergence guarantees.

Application to polynomial optimization on simple closed sets with convergence to the global minimum.

Abstract

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets $\K$ (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations.