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

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 $O(1\mathrm{/}{r}^2)$ if there exists a rational global minimizer. In this note we show that the rational minimizer condition is not necessary to obtain the $O(1\mathrm{/}{r}^2)$ bound.