An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution

We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator r (for given r). We show that the associated convergence rate is O(1/r) for quadratic polynomials. For general polynomials, if there exists a rational global minimizer over the simplex, we show that the convergence rate is also of the order O(1/r). Our results answer a question posed by De Klerk et al. [9] and improves on previously known O(1/r) bounds in the quadratic case.