# Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

## TL;DR

This paper analyzes the convergence rate of Lasserre's hierarchy for polynomial minimization on the sphere, establishing a precise Theta(1/r^2) rate and discussing implications for the generalized moment problem.

## Contribution

It provides the first exact convergence rate for Lasserre's hierarchy on the sphere, advancing understanding of its efficiency in polynomial optimization.

## Key findings

- Convergence rate is Theta(1/r^2).
- Results apply to the generalized moment problem on the sphere.
- Enhances theoretical understanding of hierarchy's efficiency.

## Abstract

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.