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arXiv · 1510.01163

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

Etienne de Klerk·Monique Laurent·Zhao Sun·Juan C. Vera
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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.

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Etienne de Klerk, Monique Laurent, Zhao Sun, Juan C. Vera. 2016-03-10. On the convergence rate of grid search for polynomial optimization over the simplex. https://arxiv.org/abs/1510.01163

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