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Polyhedral approximations of the semidefinite cone and their application

Author

Listed:
  • Yuzhu Wang

    (University of Tsukuba)

  • Akihiro Tanaka

    (Central Research Institute of Electric Power Industry)

  • Akiko Yoshise

    (University of Tsukuba)

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.

Suggested Citation

  • Yuzhu Wang & Akihiro Tanaka & Akiko Yoshise, 2021. "Polyhedral approximations of the semidefinite cone and their application," Computational Optimization and Applications, Springer, vol. 78(3), pages 893-913, April.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:3:d:10.1007_s10589-020-00255-2
    DOI: 10.1007/s10589-020-00255-2
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    References listed on IDEAS

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    1. Akihiro Tanaka & Akiko Yoshise, 2018. "LP-based tractable subcones of the semidefinite plus nonnegative cone," Annals of Operations Research, Springer, vol. 265(1), pages 155-182, June.
    2. Naohiko Arima & Sunyoung Kim & Masakazu Kojima & Kim-Chuan Toh, 2017. "A robust Lagrangian-DNN method for a class of quadratic optimization problems," Computational Optimization and Applications, Springer, vol. 66(3), pages 453-479, April.
    3. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
    4. Ken Kobayashi & Yuich Takano, 2020. "A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems," Computational Optimization and Applications, Springer, vol. 75(2), pages 493-513, March.
    5. G. Dantzig & R. Fulkerson & S. Johnson, 1954. "Solution of a Large-Scale Traveling-Salesman Problem," Operations Research, INFORMS, vol. 2(4), pages 393-410, November.
    6. Hayato Waki & Masakazu Muramatsu, 2013. "Facial Reduction Algorithms for Conic Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 188-215, July.
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