Rank conditions for exactness of semidefinite relaxations in polynomial optimization
Résumé
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 φ n and a semi-algebraic set K ⊂ 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 φ 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.
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