A new look at nonnegativity on closed sets and polynomial optimization

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 $dν=fdμ$ with support equal to K, are all positive semidefinite (if $K$ is compact $μ$ 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.