Exploiting aggregate sparsity in second-order cone relaxations for quadratic constrained quadratic programming problems

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for nonconvex quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. [Exploiting sparsity in semidefin...

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Bibliographic Details
Published in:Optimization methods & software Vol. 37; no. 2; pp. 753 - 771
Main Authors: Sheen, Heejune, Yamashita, Makoto
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 04.03.2022
Taylor & Francis Ltd
Subjects:
aggregate sparsity
chordal sparsity
Computational efficiency
Computing time
Mathematical analysis
Matrices (mathematics)
Quadratic constrained quadratic programming
Quadratic programming
second-order cone programming
Semidefinite programming
Sparsity
ISSN:1055-6788, 1029-4937
Online Access:Get full text
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Summary:Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for nonconvex quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. [Exploiting sparsity in semidefinite programming via matrix completion I: General framework, SIAM J. Optim. 11(3) (2001), pp. 647-674] and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP relaxation of nonconvex 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, nonconvex 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.
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ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2020.1827256