Type of Publication:  Journal article 
Publication status:  Published 
Author:  Lopez Quijorna, Maria 
Year of publication:  2021 
Published in:  Journal of Global Optimization ; 2021.  Springer Science.  ISSN 09255001.  eISSN 15732916 
DOI (citable link):  https://dx.doi.org/10.1007/s10898020009879 
Summary: 
A basic closed semialgebraic subset of Rn is defined by simultaneous polynomial inequalities p1≥0,…,pm≥0. 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.

Subject (DDC):  510 Mathematics 
Comment on publication:  Preprint in KOPS veröffentlicht: http://kops.unikonstanz.de/handle/123456789/38689 
Refereed:  Yes 
Online First: Journal articles that are published online before they appear as an actual part of a journal issue. Online first articles are published on the journal's website in the publisher's version.  
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LOPEZ QUIJORNA, Maria, 2021. Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction. In: Journal of Global Optimization. Springer Science. ISSN 09255001. eISSN 15732916. Available under: doi: 10.1007/s10898020009879
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