Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting

TL;DR

This paper introduces a novel proximal algorithm based on primal-dual hybrid gradient methods that exploits low-rank structures to significantly improve the scalability of semidefinite programming solvers for large instances.

Contribution

It proposes a new low-rank exploiting modification to an operator splitting method, enabling efficient large-scale semidefinite programming without extra slack variables or solving linear systems.

Findings

Achieves substantial speedup in solving large SDP instances

Effectively exploits low-rank properties for scalability

Provides convergence guarantees and open-source implementation

Abstract

In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The proposed methodology, based on the primal-dual hybrid gradient method, allows the presence of linear inequalities without the need of adding extra slack variables and avoids solving a linear system at each iteration. More importantly, it does simultaneously compute the dual variables associated with the linear constraints. The main contribution of this work is to achieve a substantial speedup by effectively adjusting the proposed algorithm in order to exploit the low-rank property inherent to several semidefinite programming problems. This proposed modification is the key element that allows the operator splitting method to efficiently scale to larger instances. Convergence guarantees are presented along with an intuitive interpretation of the algorithm. Additionally, an open source semidefinite programming solver, called ProxSDP, is made available and implementation details are discussed. Case studies are presented in order to evaluate the performance of the proposed methodology.