Type of Publication:  Dissertation 
Publication status:  Published 
URI (citable link):  http://nbnresolving.de/urn:nbn:de:bsz:35221x895ec44tgst0 
Author:  Lopez Quijorna, Maria 
Year of publication:  2018 
Summary: 
We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over a basic closed semialgebraic set, that is a polynomial optimization problem. These relaxations give an increasing sequence of lower bounds of the infimum. In this work 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 that a modified version of an optimal solution of the Lasserre relaxation is a generalized Hankel matrix. This certificate is more general than the already known certificate of an optimal solution being flat. In case that this modified matrix has a generalized Hankel form we will extract the potential minimizers with a truncated version of the GelfandNaimarkSegal 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 bring us to prove 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. We give a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem. Finally, we provide an experimental algorithm that in many cases gives us an upper bound of the global infimum of a real polynomial on R n . The algorithm that we present involves to solve a series of semidefinite programs whose feasible set is included in the feasible set of a moment relaxation. Our additional constraint try to provoke a flatness condition, like used by Curto and Fialkow, for the computed moments. At the end we present numerical results of the application of the algorithm to nonnegative polynomials which are not sums of squares. We also provide numerical results for the application of a version of this algorithm based on the method proposed by Nie, Demmel and Sturmfels.

Examination date (for dissertations):  Feb 8, 2018 
Dissertation note:  Doctoral dissertation, University of Konstanz 
Subject (DDC):  510 Mathematics 
Keywords:  Polynomial optimization, truncated moment problem 
Link to License:  In Copyright 
Bibliography of Konstanz:  Yes 
LOPEZ QUIJORNA, Maria, 2018. Detecting optimality and extracting optimal solutions in polynomial optimization based on the Lasserre relaxation [Dissertation]. Konstanz: University of Konstanz
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