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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.12359 |
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| _version_ | 1866916504880021504 |
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| author | Hu, Xiaomeng Klep, Igor Nie, Jiawang |
| author_facet | Hu, Xiaomeng Klep, Igor Nie, Jiawang |
| contents | This paper studies Positivstellensätze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ 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 $ν$ with $supp(ν) = Q$, 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 $QM(g,ν)$ associated to $(g,ν)$, under the archimedeanness assumption on $QM(g,ν)$. Here, $QM(g,ν)$ denotes the quadratic module of polynomials in $x$ that can be represented as \[τ_0(x) + \int τ_1(x,y)g_1(x, y)\, dν(y) + \cdots + \int τ_s(x,y) g_s(x, y)\, dν(y), \] where each $τ_j$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positivstellensätze and Moment problems with Universal Quantifiers Hu, Xiaomeng Klep, Igor Nie, Jiawang Optimization and Control Functional Analysis 13J30, 44A60, 90C23, 47A57, 90C34 This paper studies Positivstellensätze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ 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 $ν$ with $supp(ν) = Q$, 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 $QM(g,ν)$ associated to $(g,ν)$, under the archimedeanness assumption on $QM(g,ν)$. Here, $QM(g,ν)$ denotes the quadratic module of polynomials in $x$ that can be represented as \[τ_0(x) + \int τ_1(x,y)g_1(x, y)\, dν(y) + \cdots + \int τ_s(x,y) g_s(x, y)\, dν(y), \] where each $τ_j$ 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. |
| title | Positivstellensätze and Moment problems with Universal Quantifiers |
| topic | Optimization and Control Functional Analysis 13J30, 44A60, 90C23, 47A57, 90C34 |
| url | https://arxiv.org/abs/2401.12359 |