Positivstellensätze and Moment Problems with Universal Quantifiers

This paper studies Positivstellensätze 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 polynomial f 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. Funding: X. Hu and J. Nie are partially supported by the NSF [Grant DMS-2110780]. I. Klep is supported by the Slovenian Research Agency program P1-0222 [also Grants J1-50002, J1-60011, J1-50001, J1-2453, N1-0217, and J1-3004] and was partially supported by the Marsden Fund Council of the Royal Society of New Zealand. I. Klep’s work was partly performed within the project COMPUTE, funded within the QuantERA II program that has received funding from the EU’s H2020 research and innovation program under the GA No 101017733.