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

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.

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Correspondence to Kazuo Murota.

Additional information

The first draft of this paper appeared as the technical report “A numerical algorithm for block-diagonal decomposition of matrix \({*}\)-algebras,” issued in September 2007 as METR 2007-52, Department of Mathematical Informatics, University of Tokyo, and also as Research Report B-445, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology.

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Murota, K., Kanno, Y., Kojima, M. et al. A numerical algorithm for block-diagonal decomposition of matrix \({*}\)-algebras with application to semidefinite programming. Japan J. Indust. Appl. Math. 27, 125–160 (2010). https://doi.org/10.1007/s13160-010-0006-9

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  • DOI: https://doi.org/10.1007/s13160-010-0006-9

Keywords

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