# Comparison of Lasserre's measure--based bounds for polynomial optimization to bounds obtained by simulated annealing

## TL;DR

This paper compares Lasserre's measure-based bounds for polynomial optimization with bounds from simulated annealing, showing that Lasserre's hierarchy converges faster for polynomial functions over convex sets.

## Contribution

The paper demonstrates that for polynomial functions over convex sets, Lasserre's hierarchy provides a faster convergence rate than previously established, compared to simulated annealing bounds.

## Key findings

- Lasserre's bounds outperform simulated annealing in convergence speed.
- Faster convergence rate established for polynomial optimization over convex bodies.
- Comparison highlights advantages of measure-based bounds over stochastic methods.

## Abstract

Comparison of Lasserre's measure--based bounds for polynomial optimization to bounds obtained by simulated annealing. We consider the problem of minimizing a continuous function $f$ over a compact set $\mathbf{K}$. We compare the hierarchy of upper bounds proposed by Lasserre in [{\em SIAM J. Optim.} $21(3)$ $(2011)$, pp. $864-885$] to bounds that may be obtained from simulated annealing. We show that, when $f$ is a polynomial and $\mathbf{K}$ a convex body, this comparison yields a faster rate of convergence of the Lasserre hierarchy than what was previously known in the literature.