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

TL;DR

This paper investigates the convergence rate of grid search methods for polynomial optimization over the simplex, demonstrating that the known $O(1/r^2)$ accuracy bound holds even without the assumption of a rational global minimizer.

Contribution

It proves that the $O(1/r^2)$ convergence rate applies generally, removing the previous requirement of a rational global minimizer for polynomial optimization over the simplex.

Findings

The $O(1/r^2)$ accuracy bound holds without the rational minimizer assumption.

The convergence rate applies to polynomial optimization over the simplex.

The result extends previous error analysis to more general cases.

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.