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

Computational Optimization and Applications volume 75, pages493–513(2020)Cite this article

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.

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Notes

  1. 1.

    http://www.opt.tu-darmstadt.de/scipsdp/

  2. 2.

    https://yalmip.github.io/solver/cutsdp/

  3. 3.

    https://neos-server.org/neos/

  4. 4.

    http://scip.zib.de/

  5. 5.

    http://www.mcs.anl.gov/hs/software/DSDP/

  6. 6.

    http://www.gurobi.com/

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Acknowledgements

The authors would like to thank Mirai Tanaka for valuable comments on MISDO formulations.

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Affiliations

  1. Artificial Intelligence Laboratory, Fujitsu Laboratories LTD., 4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki-shi, Kanagawa, 211-8588, Japan

    Ken Kobayashi

  2. Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba-shi, Ibaraki, 305-8577, Japan

    Yuich Takano

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  1. Ken Kobayashi
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  2. Yuich Takano
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Correspondence to Ken Kobayashi.

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Kobayashi, K., Takano, Y. A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems. Comput Optim Appl 75, 493–513 (2020). https://doi.org/10.1007/s10589-019-00153-2

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Keywords

  • Mixed-integer optimization
  • Semidefinite optimization
  • Cutting-plane algorithm
  • Branch-and-cut algorithm