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^2 ) 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^2 ). 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.
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S.I.A.M.
SIAM Journal on Optimization
Approximation Algorithms, Quantum Information and Semidefinite Optimization
Networks and Optimization

de Klerk, E, Laurent, M, & Sun, Z. (2015). An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. SIAM Journal on Optimization, 25(3), 1498–1514.