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.

在许多提高非凸二次约束二次规划问题 (QCQP) 的半定规划 (SDP) 松弛计算效率的方法中，Fukuda等人利用了 SDP 中数据矩阵的聚合稀疏性。 [通过矩阵完成在半定规划中利用稀疏性 I：通用框架，SIAM J. Optim。11(3) (2001), pp. 647–674] 和二阶锥规划 (SOCP) 松弛已经很流行。在本文中，我们利用非凸 QCQP 的 SOCP 松弛的总体稀疏性。具体来说，我们证明了利用聚合稀疏性减少了 SOCP 松弛中二阶锥的数量，并且我们可以简化 Fukuda等人的矩阵完成过程。 在 SOCP 松弛问题的原始和对偶中，都不会失去最大行列式性质。对于数值实验，来自晶格图和池化问题的非凸 QCQP 被测试，因为它们的 SOCP 松弛提供了与 SDP 松弛相同的最优值。我们证明，对于与 SDP 松弛相同的目标值，利用聚合稀疏性可以提高 SOCP 松弛的计算效率，因此所提出的 SOCP 松弛可以处理比 SDP 松弛更大的问题。