Revisiting the convergence rate of the Lasserre hierarchy for polynomial optimization over the hypercube

TL;DR

This paper analyzes the convergence rate of Lasserre's hierarchy for polynomial optimization over the hypercube, showing an upper bound of order 1/r and discussing potential lower bounds of order 1/r^2 using the polynomial kernel method.

Contribution

The paper provides a new analysis of the convergence rate using the polynomial kernel method and discusses its limitations, offering insights into the hierarchy's efficiency.

Findings

Convergence rate is roughly 1/r using the polynomial kernel approach.

Limitations of the polynomial kernel method suggest a possible lower bound of 1/r^2.

Comparison with previous results indicates similar convergence behavior.

Abstract

We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the minimum value $f^*$, where $r$ refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter $r$. The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order $1/r$. Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order $1/r^2$ for our approach.