Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials

TL;DR

This paper proves that the sum-of-squares hierarchy for optimizing multivariate trigonometric polynomials converges at a rate of O(1/s^2) generally, and exponentially fast under certain conditions, with implications for polynomial optimization.

Contribution

It establishes the first explicit convergence rates for the sum-of-squares hierarchy applied to multivariate trigonometric polynomials, including exponential convergence under specific conditions.

Findings

Convergence rate of O(1/s^2) for general trigonometric polynomials.

Exponential convergence rate when the polynomial has finitely many minimizers with invertible Hessians.

Results extend to regular multivariate polynomials on the hypercube.

Abstract

We consider the unconstrained optimization of multivariate trigonometric polynomials by the sum-of-squares hierarchy of lower bounds. We first show a convergence rate of $O(1/s^2)$ for the relaxation with degree $s$ without any assumption on the trigonometric polynomial to minimize. Second, when the polynomial has a finite number of global minimizers with invertible Hessians at these minimizers, we show an exponential convergence rate with explicit constants. Our results also apply to minimizing regular multivariate polynomials on the hypercube.