Positivstellensatze and Moment Problems with Universal Quantifiers

• 成果类型: Article; Early Access
• 署名作者: Hu, Xiaomeng; Klep, Igor; Nie, Tiawang
• 署名单位: University of California System; University of California San Diego; University of Ljubljana; University of Primorska
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2024.0402
• 发表日期: 2025
• 关键词: squares; algorithm; sums; optimization; POLYNOMIALS; THEOREM

摘要

This paper studies Positivstellensatze and moment problems for sets K that are given by universal quantifiers. Let Q be the closed set of universal quantifiers. Fix a finite nonnegative Borel measure whose support is Q and assume it satisfies the multivariate Carleman condition. First, we prove a Positivstellensatz with universal quantifiers: if a polynomialf is positive on K, then f belongs to the associated quadratic module, under the archimedeanness assumption. Second, we prove some necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in K. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set K. Third, we present applications of the above Positivstellensatz and moment problems in semi-infinite optimization, where feasible sets are given by infinitely many constraints with universal quantifiers. This results in a new hierarchy of Moment-SOS relaxations. Its convergence is shown under some usual assumptions. The quantifier set Q is allowed to be non-semialgebraic, which makes it possible to solve some optimization problems with non-semialgebraic constraints.