Convergence rates for polynomial optimization on set products

TL;DR

This paper establishes convergence rates for polynomial optimization hierarchies on product sets, especially bi-spheres, using polynomial kernel methods, and extends results to general sphere products and quantum Wasserstein distance approximation.

Contribution

It provides the first explicit convergence rate of $O(1/t^2)$ for Schm"udgen-type hierarchies on bi-spheres and generalizes these results to arbitrary sphere products.

Findings

Convergence rate of $O(1/t^2)$ for bi-sphere polynomial optimization hierarchies.

Extension of convergence rate results to general sphere products.

Application of these results to quantum Wasserstein distance approximation.

Abstract

We consider polynomial optimization problems on Cartesian products of basic compact semialgebraic sets. The solution of such problems can be approximated as closely as desired by hierarchies of semidefinite programming relaxations, based on classical sums of squares certificates due to Putinar and Schm\"udgen. When the feasible set is the bi-sphere, i.e., the Cartesian product of two unit spheres, we show that the hierarchies based on the Schm\"udgen-type certificates converge to the global minimum of the objective polynomial at a rate in $O(1/t^2)$, where $t$ is the relaxation order. Our proof is based on the polynomial kernel method. We extend this result to arbitrary sphere products and give a general recipe to obtain convergence rates for polynomial optimization over products of distinct sets. Eventually, we rely on our results for the bi-sphere to analyze the speed of convergence of a semidefinite programming hierarchy approximating the order $2$ quantum Wasserstein distance.