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

## TL;DR

This paper extends the convergence rate analysis of Lasserre's polynomial hierarchy for minimizing polynomials over compact sets, achieving faster error bounds for broader classes of sets and measures.

## Contribution

It provides new convergence rate bounds for Lasserre's hierarchy on convex bodies and with various reference measures, improving previous estimates.

## Key findings

- Convergence rate of O(1/r^2) for hypercube with Chebyshev measure.
- Error estimate of O(log r / r) for convex bodies under certain conditions.
- Improved error bounds of O(log^2 r / r^2) for convex bodies with Lebesgue measure.

## 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 $\int_K f d\nu$. 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.