A refined error analysis for fixed-degree polynomial optimization over the simplex

TL;DR

This paper refines error bounds for polynomial optimization over the simplex, improving understanding of approximation quality for fixed-degree polynomials, especially in quadratic and cubic cases.

Contribution

It provides refined upper bounds for the difference between grid-based upper bounds and Pólya-based lower bounds in polynomial optimization over the simplex.

Findings

Refined bounds for quadratic and cubic cases.

Asymptotic refinement of bounds in the general case.

Improved understanding of approximation errors in polynomial optimization.

Abstract

We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we revisit a known upper bound obtained by taking the minimum value on a regular grid, and a known lower bound based on P\'olya's representation theorem. More precisely, we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values. Our results refine the known upper bounds in the quadratic and cubic cases, and they asymptotically refine the known upper bound in the general case.