Exploiting Aggregate Sparsity in Second Order Cone Relaxations for Quadratic Constrained Quadratic Programming Problems

TL;DR

This paper demonstrates that exploiting aggregate sparsity in second-order cone programming relaxations of quadratic constrained quadratic programming problems reduces complexity and enhances computational efficiency, enabling larger problem instances to be solved.

Contribution

It proves that aggregate sparsity reduces the number of cones in SOCP relaxations and simplifies matrix completion without losing optimality, improving efficiency over SDP relaxations.

Findings

Exploiting sparsity reduces the number of cones in SOCP relaxations.

The simplified matrix completion maintains the max-determinant property.

Larger problems can be solved efficiently with the proposed method.

Abstract

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. (2001) and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP relaxation of QCQPs. Specifically, we prove that exploiting the aggregate sparsity reduces the number of second-order cones in the SOCP relaxation, and that we can simplify the matrix completion procedure by Fukuda et al. in both primal and dual of the SOCP relaxation problem without losing the max-determinant property. For numerical experiments, QCQPs from the lattice graph and pooling problem are tested as their SOCP relaxations provide the same optimal value as the SDP relaxations. We demonstrate that exploiting the aggregate sparsity improves the computational efficiency of the SOCP relaxation for the same objective value as the SDP relaxation, thus much larger problems can be handled by the proposed SOCP relaxation than the SDP relaxation.