Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/287415 
Year of Publication: 
2021
Citation: 
[Journal:] Journal of Global Optimization [ISSN:] 1573-2916 [Volume:] 81 [Issue:] 3 [Publisher:] Springer US [Place:] New York, NY [Year:] 2021 [Pages:] 559-598
Publisher: 
Springer US, New York, NY
Abstract: 
A basic closed semialgebraic subset of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}n}$$\end{document} is defined by simultaneous polynomial inequalities p1≥0,…,pm≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{1}\ge 0,\ldots ,p_{m}\ge 0$$\end{document}. We consider Lasserre's relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality we will extract the potential minimizers with a truncated version of the Gelfand–Naimark–Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will enable us to prove, with the use of the GNS truncated construction, a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.
Subjects: 
Moment relaxation
Lassere relaxation
Polynomial optimization
Semidefinite programming
Quadrature
Truncated moment problem
GNS construction
Primary: 90C22
C26
Secondary: 44A60
D32
Persistent Identifier of the first edition: 
Creative Commons License: 
cc-by Logo
Document Type: 
Article
Document Version: 
Published Version

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