Polyhedral approximations of the semidefinite cone and their application

Yuzhu Wang (), Akihiro Tanaka () and Akiko Yoshise ()
Additional contact information
Yuzhu Wang: University of Tsukuba
Akihiro Tanaka: Central Research Institute of Electric Power Industry
Akiko Yoshise: University of Tsukuba

Computational Optimization and Applications, 2021, vol. 78, issue 3, No 8, 893-913

Abstract: 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 (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.

Keywords: Semidefinite optimization problem; Conic optimization problem; Polyhedral approximation; Semidefinite basis; Expanded semidefinite basis (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-020-00255-2 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:coopap:v:78:y:2021:i:3:d:10.1007_s10589-020-00255-2

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-020-00255-2

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().