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Consistency Cuts for Dantzig-Wolfe Reformulations

Author

Listed:
  • Jens Vinther Clausen

    (Department of Technology, Management, and Economics, Technical University of Denmark, Kongens Lyngby DK-2800, Denmark)

  • Richard Lusby

    (Department of Technology, Management, and Economics, Technical University of Denmark, Kongens Lyngby DK-2800, Denmark)

  • Stefan Ropke

    (Department of Technology, Management, and Economics, Technical University of Denmark, Kongens Lyngby DK-2800, Denmark)

Abstract

This paper introduces a family of valid inequalities, which we term consistency cuts, to be applied to a Dantzig-Wolfe reformulation (or decomposition) with linking variables. We prove that these cuts ensure an integer solution to the corresponding Dantzig-Wolfe relaxation when certain criteria to the structure of the decomposition are met. We implement the cuts and use them to solve a commonly used test set of 200 instances of the temporal knapsack problem. We assess the performance with and without the cuts and compare further to CPLEX and other solution methods that have historically been used to solve the test set. By separating consistency cuts, we show that we can obtain optimal integer solutions much faster than the other methods and even solve the remaining unsolved problems in the test set. We also perform a second test on instances from the MIPLIB 2017 online library of mixed-integer programs, showing the potential of the cuts on a wider range of problems.

Suggested Citation

  • Jens Vinther Clausen & Richard Lusby & Stefan Ropke, 2022. "Consistency Cuts for Dantzig-Wolfe Reformulations," Operations Research, INFORMS, vol. 70(5), pages 2883-2905, September.
    • Handle: RePEc:inm:oropre:v:70:y:2022:i:5:p:2883-2905
    DOI: 10.1287/opre.2021.2160
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