An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization

TL;DR

This survey reviews the convergence rates of sum of squares hierarchies in polynomial optimization, highlighting recent theoretical advances and techniques used to analyze their asymptotic behavior.

Contribution

It provides a comprehensive overview of the latest results on convergence rates of sum of squares hierarchies and introduces various analytical techniques used in their study.

Findings

Summarizes key state-of-the-art results on convergence rates.

Introduces techniques from orthogonal polynomials, approximation theory, and Fourier analysis.

Offers an accessible overview for researchers new to the topic.

Abstract

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex) optimization problems. Various hierarchies of (lower and upper) bounds have been introduced, having the remarkable property that they converge asymptotically to the global minimum. These bounds exploit algebraic representations of positive polynomials in terms of sums of squares and can be computed using semidefinite optimization. Our focus lies in the performance analysis of these hierarchies of bounds, namely, in how far the bounds are from the global minimum as the degrees of the sums of squares they involve tend to infinity. We present the main state-of-the-art results and offer a gentle introductory overview over the various techniques that have been recently developed to establish them, stemming from the theory of orthogonal polynomials, approximation theory, Fourier analysis, and more.