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On the convergence rate of grid search for polynomial optimization over the simplex

Etienne de Klerk(E.deKlerk***at***uvt.nl)
Monique Laurent(M.Laurent***at***cwi.nl)
Zhao Sun(Zhao.Sun***at***polymtl.ca)
Juan Vera(j.c.veralizcano***at***uvt.nl)

Abstract: We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator ${r} \in \mathbb{N}$. It was shown in [De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. {\em SIAM J. Optim.} 25(3) 1498--1514 (2015)] that the relative accuracy of this approximation depends on $r$ as $O(1/r^2)$ if there exists a rational global minimizer. In this note we show that the rational minimizer condition is not necessary to obtain the $O(1/r^2)$ bound.

Keywords: Polynomial optimization, Taylor's theorem

Category 1: Global Optimization (Theory )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: Technical report, Tilburg University, CWI Amsterdam, and Polytechnique Montreal, October 2015.

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Entry Submitted: 10/05/2015
Entry Accepted: 10/05/2015
Entry Last Modified: 10/05/2015

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