An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution

TL;DR

This paper analyzes the convergence rate of rational approximation methods for polynomial optimization over the simplex, showing an improved rate of O(1/r^2) for quadratic polynomials and cases with rational minimizers.

Contribution

It establishes a faster convergence rate for rational grid approximations in polynomial optimization over the simplex, answering a question from prior research.

Findings

Convergence rate is O(1/r^2) for quadratic polynomials.

Rate is also O(1/r^2) if a rational global minimizer exists.

Improves previous bounds of O(1/r) in the quadratic case.

Abstract

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. (2013) and improves on previously known $O(1/r)$ bounds in the quadratic case.