Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel

TL;DR

This paper studies sum-of-squares hierarchies for polynomial optimization over compact sets, establishing convergence rates and connecting these hierarchies to the Christoffel-Darboux kernel, with implications for optimization efficiency.

Contribution

It demonstrates convergence rates for sum-of-squares hierarchies on specific sets and links these hierarchies to the Christoffel-Darboux kernel, providing new analytical tools.

Findings

Hierarchies converge at rate O(1/r^2) for the unit ball and simplex.

Established a connection between Lasserre's hierarchies and the Christoffel-Darboux kernel.

Utilized closed-form expressions for the kernel to analyze convergence.

Abstract

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set ${\mathbf{X} \subseteq \mathbb{R}^n}$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schm\"udgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schm\"udgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel-Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.