Exploiting aggregate sparsity in second-order cone relaxations for quadratic constrained quadratic programming problems. (4th March 2022)

• Record Type: Journal Article
• Authors: Sheen, Heejune; Yamashita, Makoto
• Is Part Of: Optimization methods and software. Volume 37:Number 2(2022)
• Page Start: 753
• Page End: 771
• Publication Date: 2022-03-04
• Subjects: Quadratic constrained quadratic programming -- semidefinite programming -- second-order cone programming -- aggregate sparsity -- chordal sparsity; 90C20 -- 90C22 -- 90C25 -- 90C26; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2020.1827256
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : 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.