Rank conditions for exactness of semidefinite relaxations in polynomial optimization

• 成果类型: Article; Early Access
• 署名作者: Lasserre, Jean B.
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Communaute d'universites et etablissements de Toulouse (Comue); Universite Toulouse 1 Capitole; Toulouse School of Economics
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02417-2
• 发表日期: 2026-08-21
• 关键词: Truncated -moment problem; polynomial optimization; moment-sos hierarchy; Exactness of relaxations; global optimization; hierarchy

摘要

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 of Rd . 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 n-v+1 , then phi n restricted to degree- 2n-1 moments, 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.