Leveraging Christoffel-Darboux Kernels to Strengthen Moment-SOS Relaxations

TL;DR

This paper introduces a novel method that uses Christoffel-Darboux kernels to improve the accuracy of low-order Moment-SOS relaxations, making polynomial optimization more computationally feasible and effective.

Contribution

It presents a new approach that enhances low-order relaxations with Christoffel-Darboux kernels, improving bounds and minimizer extraction in polynomial optimization.

Findings

Strengthened relaxations yield significantly better bounds.

Method facilitates minimizer extraction in quadratic programs.

Approach is effective on various quadratically constrained problems.

Abstract

The classical Moment-Sum Of Squares hierarchy allows to approximate a global minimum of a polynomial optimization problem through semidefinite relaxations of increasing size. However, for many optimization instances, solving higher order relaxations becomes impractical or even impossible due to the substantial computational demands they impose. To address this, existing methods often exploit intrinsic problem properties, such as symmetries or sparsity. Here, we present a complementary approach, which enhances the accuracy of computationally more efficient low-order relaxations by leveraging Christoffel-Darboux kernels. Such strengthened relaxations often yield significantly improved bounds or even facilitate minimizer extraction. We illustrate the efficiency of our approach on several classes of important quadratically constrained quadratic Programs.