Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube

TL;DR

This paper analyzes the convergence rates of Lasserre's hierarchy for polynomial optimization on the hypercube, providing matching lower bounds and revealing the influence of orthogonal polynomial zeroes.

Contribution

It offers a refined convergence analysis and establishes tight bounds for the hierarchy's rate on specific examples, linking it to orthogonal polynomial properties.

Findings

Convergence rates are determined by extremal zeroes of orthogonal polynomials.

Lower bounds match upper bounds, confirming the true convergence rate.

Results apply to a class of polynomial optimization problems on the hypercube.

Abstract

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.