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 a closed set and let $g=(g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_s)$ 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.