Mathematical Programming

, Volume 95, Issue 2, pp 303–327 | Cite as

Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results

  • Kazuhide Nakata
  • Katsuki Fujisawa
  • Mituhiro Fukuda
  • Masakazu Kojima
  • Kazuo Murota

Abstract.

 In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Kazuhide Nakata
    • 1
  • Katsuki Fujisawa
    • 2
  • Mituhiro Fukuda
    • 3
  • Masakazu Kojima
    • 3
  • Kazuo Murota
    • 4
  1. 1.Department of Industrial Engineering and Management, Tokyo Institute of Technology, 2-12-1, Oh-Okayama, Meguro-Ku, Tokyo 152-8552, Japan, e-mail: knakata@me.titech.ac.jpJP
  2. 2.Department of Architecture and Architectural Systems, Kyoto University, Kyoto 606-8501, Japan, e-mail: fujisawa@is-mj.archi.kyoto-u.ac.jpJP
  3. 3.Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1, Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan, e-mail: mituhiro@is.titech. ac.jp, kojima@is.titech.ac.jpJP
  4. 4.Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan, e-mail: murota@mist.i.u-tokyo.ac.jpJP

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