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SDP-based branch-and-bound for non-convex quadratic integer optimization

Author

Listed:
  • Christoph Buchheim

    (TU Dortmund)

  • Maribel Montenegro

    (TU Dortmund)

  • Angelika Wiegele

    (Alpen-Adria-Universität Klagenfurt)

Abstract

Semidefinite programming (SDP) relaxations have been intensively used for solving discrete quadratic optimization problems, in particular in the binary case. For the general non-convex integer case with box constraints, the branch-and-bound algorithm Q-MIST has been proposed by Buchheim and Wiegele (Math Program 141(1–2):435–452, 2013), which is based on an extension of the well-known SDP-relaxation for max-cut. For solving the resulting SDPs, Q-MIST uses an off-the-shelf interior point algorithm. In this paper, we present a tailored coordinate ascent algorithm for solving the dual problems of these SDPs. Building on related ideas of Dong (SIAM J Optim 26(3):1962–1985, 2016), it exploits the particular structure of the SDPs, most importantly a small rank of the constraint matrices. The latter allows both an exact line search and a fast incremental update of the inverse matrices involved, so that the entire algorithm can be implemented to run in quadratic time per iteration. Moreover, we describe how to extend this approach to a certain two-dimensional coordinate update. Finally, we explain how to include arbitrary linear constraints into this framework, and evaluate our algorithm experimentally.

Suggested Citation

  • Christoph Buchheim & Maribel Montenegro & Angelika Wiegele, 2019. "SDP-based branch-and-bound for non-convex quadratic integer optimization," Journal of Global Optimization, Springer, vol. 73(3), pages 485-514, March.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:3:d:10.1007_s10898-018-0717-z
    DOI: 10.1007/s10898-018-0717-z
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    References listed on IDEAS

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    1. C. Helmberg & F. Rendl & R. Weismantel, 2000. "A Semidefinite Programming Approach to the Quadratic Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 197-215, June.
    2. Samuel Burer & Dieter Vandenbussche, 2009. "Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound," Computational Optimization and Applications, Springer, vol. 43(2), pages 181-195, June.
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    Cited by:

    1. Cheng Lu & Zhibin Deng, 2021. "A branch-and-bound algorithm for solving max-k-cut problem," Journal of Global Optimization, Springer, vol. 81(2), pages 367-389, October.

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