Positivstellens\"atze and Moment problems with Universal Quantifiers

Abstract

This paper studies Positivstellens\"atze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$. Then $K$ is described as the set of all points $x$ such that each $g_j(x, y) \ge 0$ for all $y \in Q$. Fix a finite nonnegative Borel measure $\nu$ with $supp(\nu) = Q$, and assume it satisfies the multivariate Carleman condition. The first main result of the paper is a Positivstellensatz with universal quantifiers: if a polynomial $f(x)$ is positive on $K$, then it belongs to the quadratic module $QM(g,\nu)$ associated to $(g,\nu)$, under the archimedeanness assumption on $QM(g,\nu)$. Here, $QM(g,\nu)$ denotes the quadratic module of polynomials in $x$ that can be represented as \[\tau_0(x) + \int \tau_1(x,y)g_1(x, y)\, d\nu(y) + \cdots + \int \tau_s(x,y) g_s(x, y)\, d\nu(y), \] where each $\tau_j$ is a sum of squares polynomial. Second, necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in $K$ are given. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set $K$. Finally, applications of these results for solving semi-infinite optimization problems are presented.