On the convergence rate of grid search for polynomial optimization over the simplex.
We consider the approximate minimization of a given polynomial on the s tandard 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 relative accuracy of this approximation depends on r as O(1/r2) if there exists a rational global minimizer. In this note we show that th e rational minimizer condition is not necessary to obtain the O(1/r2) bound.
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de Klerk, E, Laurent, M, Sun, Z, & Vera, J.C. (2015). On the convergence rate of grid search for polynomial optimization over the simplex. .