A Sublevel Moment-SOS Hierarchy for Polynomial Optimization

TL;DR

This paper proposes a flexible sublevel Moment-SOS hierarchy for polynomial optimization that interpolates between existing relaxations, offering improved bounds and computational advantages for complex problems in combinatorial optimization and deep learning.

Contribution

It introduces a novel sublevel hierarchy that enhances the Moment-SOS approach by allowing adjustable relaxation levels, improving bounds and computational efficiency.

Findings

Lower bounds improve upon Shor's relaxation.

Significantly closer to optimal or best-known bounds.

Effective for large-scale combinatorial and deep learning problems.

Abstract

We introduce a sublevel Moment-SOS hierarchy where each SDP relaxation can be viewed as an intermediate (or interpolation) between the d-th and (d+1)-th order SDP relaxations of the Moment-SOS hierarchy (dense or sparse version). With the flexible choice of determining the size (level) and number (depth) of subsets in the SDP relaxation, one is able to obtain different improvements compared to the d-th order relaxation, based on the machine memory capacity. In particular, we provide numerical experiments for d=1 and various types of problems both in combinatorial optimization (Max-Cut, Mixed Integer Programming) and deep learning (robustness certification, Lipschitz constant of neural networks), where the standard Lasserre's relaxation (or its sparse variant) is computationally intractable. In our numerical results, the lower bounds from the sublevel relaxations improve the bound from Shor's relaxation (first order Lasserre's relaxation) and are significantly closer to the optimal value or to the best-known lower/upper bounds.