Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion

Sunyoung Kim, Masakazu Kojima, Martin Mevissen, Makoto Yamashita

Research output: Contribution to journalArticlepeer-review

56 Scopus citations

Abstract

A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity, characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the problem, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the problem. Four conversion methods are proposed in this framework: two for exploiting the d-space sparsity and the other two for exploiting the r-space sparsity. When applied to a polynomial semidefinite program (SDP), these conversion methods enhance the structured sparsity of the problem called the correlative sparsity. As a result, the resulting polynomial SDP can be solved more effectively by applying the sparse SDP relaxation. Preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems.

Original languageEnglish
Pages (from-to)33-68
Number of pages36
JournalMathematical Programming
Volume129
Issue number1
DOIs
StatePublished - Sep 2011

Keywords

  • Chordal Graph
  • Matrix Inequalities
  • Polynomial Optimization
  • Positive Semidefinite Matrix Completion
  • Semidefinite Program
  • Sparsity

Fingerprint

Dive into the research topics of 'Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion'. Together they form a unique fingerprint.

Cite this