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A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems

Author

Listed:
  • Ken Kobayashi

    (Fujitsu Laboratories LTD.)

  • Yuich Takano

    (University of Tsukuba)

Abstract

We consider a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite (psd) constraint is relaxed, and the resultant mixed-integer linear optimization problem is solved repeatedly, imposing at each iteration a valid inequality for the psd constraint. We prove the convergence properties of the algorithm. Moreover, to speed up the computation, we devise a branch-and-cut algorithm, in which valid inequalities are dynamically added during a branch-and-bound procedure. We test the computational performance of our cutting-plane and branch-and-cut algorithms for three types of MISDO problem: random instances, computing restricted isometry constants, and robust truss topology design. Our experimental results demonstrate that, for many problem instances, our branch-and-cut algorithm delivered superior performance compared with general-purpose MISDO solvers in terms of computational efficiency and stability.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:2:d:10.1007_s10589-019-00153-2
    DOI: 10.1007/s10589-019-00153-2
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    References listed on IDEAS

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    1. Adelaide Cerveira & Agostinho Agra & Fernando Bastos & Joaquim Gromicho, 2013. "A new Branch and Bound method for a discrete truss topology design problem," Computational Optimization and Applications, Springer, vol. 54(1), pages 163-187, January.
    2. Gábor Braun & Samuel Fiorini & Sebastian Pokutta & David Steurer, 2015. "Approximation Limits of Linear Programs (Beyond Hierarchies)," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 756-772, March.
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    Cited by:

    1. Ken Kobayashi & Yuichi Takano & Kazuhide Nakata, 2021. "Bilevel cutting-plane algorithm for cardinality-constrained mean-CVaR portfolio optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 493-528, October.
    2. 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.
    3. Katsuhiro Tanaka & Rei Yamamoto, 2023. "Ellipsoidal buffered area under the curve maximization model with variable selection in credit risk estimation," Computational Management Science, Springer, vol. 20(1), pages 1-28, December.

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