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.
Categories: math.OC