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Automatic Dantzig–Wolfe reformulation of mixed integer programs

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Abstract

Dantzig–Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That is, we perform a rigorous experimental study, which results in identifying a score to estimate the quality of a decomposition: after building a set of potentially good candidates, we exploit such a score to detect which decomposition might be useful for Dantzig–Wolfe reformulation of a MIP. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the automatically reformulated model using column generation. Our findings support the idea that Dantzig–Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community.

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Acknowledgments

We sincerely thank an anonymous referee for thoughtful and motivating feedback, which led to a more meaningful experimental setup and a much improved organization of the material.

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Authors and Affiliations

  1. Operations Research, RWTH Aachen University, Kackertstraße 7, 52072 , Aachen, Germany

    Martin Bergner & Marco E. Lübbecke

  2. Dipartimento di Informatica, Università degli Studi di Milano, Via Bramante 65, 26013 , Crema, Italy

    Alberto Ceselli

  3. LAMSADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 , Paris, France

    Fabio Furini

  4. DEI, Università di Bologna, Viale Risorgimento 2, 40136 , Bologna, Italy

    Alberto Caprara & Enrico Malaguti

  5. LIPN, Équipe AOC, Université Paris 13, 99 Avenue Jean-Baptiste Clément, 93430 , Villetaneuse, France

    Emiliano Traversi

Authors
  1. Martin Bergner
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  2. Alberto Caprara
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  3. Alberto Ceselli
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  4. Fabio Furini
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  5. Marco E. Lübbecke
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  6. Enrico Malaguti
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  7. Emiliano Traversi
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Corresponding author

Correspondence to Marco E. Lübbecke.

Additional information

A preliminary version of this paper appeared in [3].

Martin Bergner was supported by the German Research Foundation (DFG) as part of the Priority Program “Algorithm Engineering” under Grants No. LU770/4-1 and 4-2.

Alberto Caprara: Deceased.

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Bergner, M., Caprara, A., Ceselli, A. et al. Automatic Dantzig–Wolfe reformulation of mixed integer programs. Math. Program. 149, 391–424 (2015). https://doi.org/10.1007/s10107-014-0761-5

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  • DOI: https://doi.org/10.1007/s10107-014-0761-5

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