Rank conditions for exactness of semidefinite relaxations in polynomial optimization

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.