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Polyhedral approximations of the semidefinite cone and their application

Yuzhu Wang (wangyuzhu820***at***yahoo.co.jp)
Akihiro Tanaka (a-tanaka***at***criepi.denken.or.jp)
Akiko Yoshise (yoshise***at***sk.tsukuba.ac.jp)

Abstract: We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

Keywords: Semidefinite optimization problems; Conic optimization problems; Polyhedral approximation; Semidefinite bases; Expanded semidefinite bases.

Category 1: Linear, Cone and Semidefinite Programming

Citation: Discussion Paper Series No. 1359, Department of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan.

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Entry Submitted: 04/30/2019
Entry Accepted: 05/01/2019
Entry Last Modified: 11/27/2019

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