Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets

Abstract

We consider the problem of computing the minimum value $f_{\min ,K}$ of a polynomial $f$ over a compact set $K\subseteq {\mathbb{R}}^n$, which can be reformulated as finding a probability measure $\nu$ on $K$ minimizing ${\int }_{K}fd\nu$. Lasserre showed that it suffices to consider such measures of the form $\nu =q\mu$, where $q$ is a sum-of-squares polynomial and $\mu$ is a given Borel measure supported on $K$. By bounding the degree of $q$ by $2r$ one gets a converging hierarchy of upper bounds $f^{(r)}$ for $f_{\min ,K}$. When $K$ is the hypercube $[-1,1{]}^n$, equipped with the Chebyshev measure, the parameters $f^{(r)}$ are known to converge to $f_{\min ,K}$ at a rate in $O(1/r^2)$. We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in $O(\log r/r)$ when $K$ satisfies a minor geometrical condition, and in $O({\log }^{2}r/r^2)$ when $K$ is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $O(1/\sqrt{r})$ and $O(1/r)$ for these two respective cases.

30 pages with 10 figures. Update notes for second version: Added a new section containing numerical examples that illustrate the theoretical results -- Fixed minor mistakes/typos -- Improved some notation -- Clarified certain explanations in the text