scholarly journals Relative Perturbation Theory for Diagonally Dominant Matrices

2014 ◽  
Vol 35 (4) ◽  
pp. 1303-1328 ◽  
Author(s):  
Megan Dailey ◽  
Froilán M. Dopico ◽  
Qiang Ye
Keyword(s):  
Perturbation Theory ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Relative Perturbation Theory

scholarly journals Polyhedral approximations of the semidefinite cone and their application

2021 ◽  
Author(s):  
Yuzhu Wang ◽  
Akihiro Tanaka ◽  
Akiko Yoshise
Keyword(s):  
Initial Approximation ◽  
Semidefinite Relaxation ◽  
Stable Set ◽  
Cutting Plane Methods ◽  
Diagonally Dominant Matrices ◽  
Maximum Stable Set ◽  
Simple Expansion ◽  
Diagonally Dominant ◽  
Polyhedral Approximations ◽  
Semidefinite Cone

AbstractWe 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 (Ann Oper Res 265:155–182, 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.

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