Sparse Sub-gaussian Random Projections for Semidefinite Programming Relaxations

TL;DR

This paper introduces sparse sub-gaussian random projections to efficiently approximate semidefinite programming relaxations in combinatorial optimization, reducing computational complexity while maintaining solution quality.

Contribution

It provides theoretical bounds on projection quality and demonstrates practical effectiveness in approximating SDP relaxations for MAXCUT, MAX-2-SAT, and graph stability problems.

Findings

Effective reduction in SDP problem size for large graphs

Good approximation bounds for MAXCUT and MAX-2-SAT relaxations

Computational experiments confirm efficiency and accuracy of the approach

Abstract

Random projection, a dimensionality reduction technique, has been found useful in recent years for reducing the size of optimization problems. In this paper, we explore the use of sparse sub-gaussian random projections to approximate semidefinite programming (SDP) problems by reducing the size of matrix variables, thereby solving the original problem with much less computational effort. We provide some theoretical bounds on the quality of the projection in terms of feasibility and optimality that explicitly depend on the sparsity parameter of the projector. We investigate the performance of the approach for semidefinite relaxations appearing in polynomial optimization, with a focus on combinatorial optimization problems. In particular, we apply our method to the semidefinite relaxations of MAXCUT and MAX-2-SAT. We show that for large unweighted graphs, we can obtain a good bound by solving a projection of the semidefinite relaxation of MAXCUT. We also explore how to apply our method to find the stability number of four classes of imperfect graphs by solving a projection of the second level of the Lasserre Hierarchy. Overall, our computational experiments show that semidefinite programming problems appearing as relaxations of combinatorial optimization problems can be approximately solved using random projections as long as the number of constraints is not too large.