Moments matrices, real algebraic geometry and polynomial optimization

Marta Abril Bucero 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : The objective of this thesis is to compute the optimum of a polynomial on a closed basic semialgebraic set and the points where this optimum is reached. To achieve this goal we combine border basis method with Lasserre's hierarchy in order to reduce the size of the moment matrices in the SemiDefinite Programming (SDP) problems. In order to verify if the minimum is reached we describe a new criterion to verify the flat extension condition using border basis. Combining these new results we provide a new algorithm which computes the optimum and the minimizers points. We show several experimentations and some applications in different domains which prove the perfomance of the algorithm. Theorethically we also prove the finite convergence of a SDP hierarchie contructed from a Karush-Kuhn-Tucker ideal and its consequences in particular cases. We also solve the particular case where the minimizers are not KKT points using Fritz-John Variety.
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Marta Abril Bucero. Moments matrices, real algebraic geometry and polynomial optimization. General Mathematics [math.GM]. Université Nice Sophia Antipolis, 2014. English. ⟨NNT : 2014NICE4118⟩. ⟨tel-01130691⟩

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