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Min-Sup-Min Robust Combinatorial Optimization with Few Recourse Solutions

Author

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  • Ayşe N. Arslan

    (Université Rennes, Institut National des Sciences Appliquées de Rennes, Centre National de la Recherche Scientifique, Institut de Recherche Mathématique de Rennes, Unité Mixte de Recherche 6625, F-35000 Rennes, France)

  • Michael Poss

    (Laboratory of Computer Science, Robotics and Microelectronics of Montpellier (LIRMM), UMR CNRS 5506, University of Montpellier, 34095 Montpellier, France)

  • Marco Silva

    (Centro de Engenharia e Gestão Industrial (CEGI), Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ciência (INESC TEC), Faculdade de Engenharia da Universidade do Porto, 4200-465 Porto, Portugal)

Abstract

In this paper, we consider a variant of adaptive robust combinatorial optimization problems where the decision maker can prepare K solutions and choose the best among them upon knowledge of the true data realizations. We suppose that the uncertainty may affect the objective and the constraints through functions that are not necessarily linear. We propose a new exact algorithm for solving these problems when the feasible set of the nominal optimization problem does not contain too many good solutions. Our algorithm enumerates these good solutions, generates dynamically a set of scenarios from the uncertainty set, and assigns the solutions to the generated scenarios using a vertex p -center formulation, solved by a binary search algorithm. Our numerical results on adaptive shortest path and knapsack with conflicts problems show that our algorithm compares favorably with the methods proposed in the literature. We additionally propose a heuristic extension of our method to handle problems where it is prohibitive to enumerate all good solutions. This heuristic is shown to provide good solutions within a reasonable solution time limit on the adaptive knapsack with conflicts problem. Finally, we illustrate how our approach handles nonlinear functions on an all-or-nothing subset problem taken from the literature. Summary of Contribution: Our paper describes a new exact algorithm for solving adaptive robust combinatorial optimization problems when the feasible set of the nominal optimization problems does not contain too many good solutions. Its development relies on a progressive relaxation of the problem augmented with a row-and-column generation technique. Its efficient execution requires a reformulation of this progressive relaxation, coupled with dominance rules and a binary search algorithm. The proposed algorithm is amenable to exploiting the special structures of the problems considered as illustrated with various applications throughout the paper. A practical view is provided by the proposition of a heuristic variant. Our computational experiments show that our proposed exact solution method outperforms the existing methodologies and therefore pushes the computational envelope for the class of problems considered.

Suggested Citation

  • Ayşe N. Arslan & Michael Poss & Marco Silva, 2022. "Min-Sup-Min Robust Combinatorial Optimization with Few Recourse Solutions," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2212-2228, July.
  • Handle: RePEc:inm:orijoc:v:34:y:2022:i:4:p:2212-2228
    DOI: 10.1287/ijoc.2021.1156
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    References listed on IDEAS

    as
    1. Dimitris Bertsimas & Iain Dunning, 2016. "Multistage Robust Mixed-Integer Optimization with Adaptive Partitions," Operations Research, INFORMS, vol. 64(4), pages 980-998, August.
    2. Josette Ayoub & Michael Poss, 2016. "Decomposition for adjustable robust linear optimization subject to uncertainty polytope," Computational Management Science, Springer, vol. 13(2), pages 219-239, April.
    3. Ruslan Sadykov & François Vanderbeck, 2013. "Bin Packing with Conflicts: A Generic Branch-and-Price Algorithm," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 244-255, May.
    4. Artur Alves Pessoa & Michael Poss, 2015. "Robust Network Design with Uncertain Outsourcing Cost," INFORMS Journal on Computing, INFORMS, vol. 27(3), pages 507-524, August.
    5. Grani A. Hanasusanto & Daniel Kuhn & Wolfram Wiesemann, 2015. "K -Adaptability in Two-Stage Robust Binary Programming," Operations Research, INFORMS, vol. 63(4), pages 877-891, August.
    6. Aissi, Hassene & Bazgan, Cristina & Vanderpooten, Daniel, 2009. "Min-max and min-max regret versions of combinatorial optimization problems: A survey," European Journal of Operational Research, Elsevier, vol. 197(2), pages 427-438, September.
    7. Chassein, André & Goerigk, Marc & Kurtz, Jannis & Poss, Michael, 2019. "Faster algorithms for min-max-min robustness for combinatorial problems with budgeted uncertainty," European Journal of Operational Research, Elsevier, vol. 279(2), pages 308-319.
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