Rank conditions for exactness of semidefinite relaxations in polynomial optimization

TL;DR

This paper establishes a new rank-based condition for the exactness of semidefinite relaxations in polynomial optimization, enabling finite convergence guarantees and extraction of global minimizers for certain problems.

Contribution

It introduces a novel rank condition that ensures finite convergence of the Moment-SOS hierarchy and allows extraction of solutions in polynomial optimization.

Findings

Rank condition guarantees finite convergence of the hierarchy.

Condition applies to constrained polynomial optimization problems.

Enables extraction of global minimizers from pseudo-moment sequences.

Abstract

We consider the Moment-SOS hierarchy in polynomial optimization. We first provide a sufficient condition to solve the truncated K-moment problem associated with a given degree-$2n$ pseudo-moment sequence $\phi$ n and a semi-algebraic set $K \subset \mathbb{R}^d$. Namely, let $2v$ be the maximum degree of the polynomials that describe $K$. If the rank $r$ of its associated moment matrix is less than $nv + 1$, then $\phi^n$ has an atomic representing measure supported on at most $r$ points of $K$. When used at step-$n$ of the Moment-SOS hierarchy, it provides a sufficient condition to guarantee its finite convergence (i.e., the optimal value of the corresponding degree-n semidefinite relaxation of the hierarchy is the global minimum). For Quadratic Constrained Quadratic Problems (QCQPs) one may also recover global minimizers from the optimal pseudo-moment sequence. Our condition is in the spirit of Blekherman's rank condition and while on the one-hand it is more restrictive, on the other hand it applies to constrained POPs as it provides a localization on $K$ for the representing measure.