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 ∈ 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. SIAM J. Optim. 25(3) 1498–1514 (2015)] that the accuracy of this approximation depends on r as O(1/r) 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) bound.