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A numerical algorithm for block-diagonal decomposition of matrix \({*}\)-algebras with application to semidefinite programming

  • Kazuo Murota
  • Yoshihiro Kanno
  • Masakazu Kojima
  • Sadayoshi Kojima
Original Paper Area 2

Abstract

Motivated by recent interest in group-symmetry in semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of matrices, or equivalently the irreducible decomposition of the generated matrix \({*}\)-algebra. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. The main issues of the proposed approach are presented in this paper under some assumptions, while the companion paper gives an algorithm with full generality. Numerical examples of truss and frame designs are also presented.

Keywords

Matrix \({*}\)-algebra Block-diagonalization Group symmetry Sparsity Semidefinite programming 

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Copyright information

© The JJIAM Publishing Committee and Springer 2010

Authors and Affiliations

  • Kazuo Murota
    • 1
  • Yoshihiro Kanno
    • 1
  • Masakazu Kojima
    • 2
  • Sadayoshi Kojima
    • 2
  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan

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