Instance-specific linear relaxations of semidefinite optimization problems

TL;DR

This paper presents a novel technique for deriving linear relaxations of semidefinite programs with guarantees, improving approximation quality for problems like max cut and Lovasz theta.

Contribution

It introduces a generic, commutativity-based method to create effective linear relaxations of SDPs, with theoretical guarantees and practical applications.

Findings

Linear relaxations match the SDP optimal value under certain conditions.

The proposed LP for max cut is stronger than existing bounds.

Experiments demonstrate improved warm-starting and approximation for various SDPs.

Abstract

We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of the SDP and the proposed linear relaxation match, which we then relax to provide a flexible methodology to derive effective linear relaxations. We specialize these results to provide linear programs that approximate well-known semidefinite programs for the max cut problem proposed by Poljak and Rendl, and the Lovasz theta number; we prove that the linear program proposed for max cut certifies a known eigenvalue bound for the maximum cut value and is in fact stronger. Our ideas can be used to warm-start algorithms that solve semidefinite programs by iterative polyhedral approximation of the feasible region. We verify this capability through multiple experiments on the max cut semidefinite program, the Lovasz theta number and on three families of semidefinite programs obtained as convex relaxations of certain quadratically constrained quadratic problems.