A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications

TL;DR

This paper introduces a perturbed sums of squares theorem for polynomial optimization, leading to new, smaller SDP relaxations that approximate the optimal value within a controlled error margin.

Contribution

The paper develops a novel perturbed sums of squares theorem and proposes smaller SDP relaxations for polynomial optimization with provable bounds.

Findings

SDP relaxations are often significantly smaller in dimension.

Relaxations provide bounds within epsilon of the true optimal value.

Applications demonstrate practical effectiveness of the new relaxations.

Abstract

We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^\ast - \epsilon)$ and from above by $f^\ast + \epsilon (n+1)$, where $f^\ast$ is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.