Dealing with inequality constraints in large-scale semidefinite relaxations for graph coloring and maximum clique problems

TL;DR

This paper extends an ADMM algorithm to handle inequality constraints in large-scale semidefinite relaxations, demonstrating its effectiveness on combinatorial problems like graph coloring and maximum clique, with favorable results compared to existing solvers.

Contribution

The paper introduces an ADMM extension for SDPs with inequalities and shows its practical efficiency on large-scale combinatorial problem relaxations.

Findings

The extended ADMM compares favorably with SDPNAL+ on random instances.

Even approximate dual solutions can yield valid bounds after post-processing.

The method effectively handles large-scale SDP relaxations of classical problems.

Abstract

Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods. However, when the dimension of the problem gets large, interior point methods become impractical in terms of both computational time and memory requirements. Certain first-order methods, such as Alternating Direction Methods of Multipliers (ADMMs), established as suitable algorithms to deal with large-scale SDPs and gained growing attention over the past decade. In this paper, we focus on an ADMM designed for SDPs in standard form and extend it to deal with inequalities when solving SDPs in general form. Beside numerical results on randomly generated instances, where we show that our method compares favorably with respect to the state-of-the-art solver SDPNAL+, we present results on instances from SDP relaxations of classical combinatorial problems such as the graph coloring problem and the maximum clique problem. Through extensive numerical experiments, we show that even an inaccurate dual solution, obtained at a generic iteration of our proposed ADMM, can represent an efficiently recovered valid bound on the optimal solution of the combinatorial problems considered, as long as an appropriate post-processing procedure is applied.