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An oracle-based framework for robust combinatorial optimization

Author

Listed:
  • Enrico Bettiol

    (TU Dortmund)

  • Christoph Buchheim

    (TU Dortmund)

  • Marianna De Santis

    (Sapienza University of Rome)

  • Francesco Rinaldi

    (University of Padova)

Abstract

We propose a general solution approach for min-max-robust counterparts of combinatorial optimization problems with uncertain linear objectives. We focus on the discrete scenario case, but our approach can be extended to other types of uncertainty sets such as polytopes or ellipsoids. Concerning the underlying certain problem, the algorithm is entirely oracle-based, i.e., our approach only requires a (primal) algorithm for solving the certain problem. It is thus particularly useful in case the certain problem is well-studied but its combinatorial structure cannot be directly exploited in a tailored robust optimization approach, or in situations where the underlying problem is only defined implicitly by a given software. The idea of our algorithm is to solve the convex relaxation of the robust problem by a simplicial decomposition approach, the main challenge being the non-differentiability of the objective function in the case of discrete or polytopal uncertainty. The resulting dual bounds are then used within a tailored branch-and-bound framework for solving the robust problem to optimality. By a computational evaluation, we show that our method outperforms straightforward linearization approaches on the robust minimum spanning tree problem. Moreover, using the Concorde solver for the certain oracle, our approach computes much better dual bounds for the robust traveling salesman problem in the same amount of time.

Suggested Citation

  • Enrico Bettiol & Christoph Buchheim & Marianna De Santis & Francesco Rinaldi, 2024. "An oracle-based framework for robust combinatorial optimization," Journal of Global Optimization, Springer, vol. 88(1), pages 27-51, January.
  • Handle: RePEc:spr:jglopt:v:88:y:2024:i:1:d:10.1007_s10898-023-01271-2
    DOI: 10.1007/s10898-023-01271-2
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    References listed on IDEAS

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    1. Christoph Buchheim & Marianna De Santis & Francesco Rinaldi & Long Trieu, 2018. "A Frank–Wolfe based branch-and-bound algorithm for mean-risk optimization," Journal of Global Optimization, Springer, vol. 70(3), pages 625-644, March.
    2. Christoph Buchheim & Jannis Kurtz, 2018. "Robust combinatorial optimization under convex and discrete cost uncertainty," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(3), pages 211-238, September.
    3. Torbjörn Larsson & Michael Patriksson, 1992. "Simplicial Decomposition with Disaggregated Representation for the Traffic Assignment Problem," Transportation Science, INFORMS, vol. 26(1), pages 4-17, February.
    4. Enrico Bettiol & Lucas Létocart & Francesco Rinaldi & Emiliano Traversi, 2020. "A conjugate direction based simplicial decomposition framework for solving a specific class of dense convex quadratic programs," Computational Optimization and Applications, Springer, vol. 75(2), pages 321-360, March.
    5. Nicolas Kämmerling & Jannis Kurtz, 2020. "Oracle-based algorithms for binary two-stage robust optimization," Computational Optimization and Applications, Springer, vol. 77(2), pages 539-569, November.
    Full references (including those not matched with items on IDEAS)

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