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Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube

Etienne De Klerk(e.deklerk***at***uvt.nl)
Monique Laurent(m.laurent***at***uvt.nl)

Abstract: We consider the problem of minimizing a polynomial on the hypercube [0,1]^n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmuedgen (1991). The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the hypercube.

Keywords: Positivstellensatz, positive polynomial, sum of squares of polynomials, bound constrained optimization of polynomials, multivariate Bernstein approximation, semidefinite programming

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Global Optimization

Citation: Preprint, Tilburg University, The Netherlands, April 2010

Download: [PDF]

Entry Submitted: 04/09/2010
Entry Accepted: 04/09/2010
Entry Last Modified: 04/09/2010

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