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Reduce-and-Split Cuts: Improving the Performance of Mixed-Integer Gomory Cuts

Author

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  • Kent Andersen

    (CORE, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium)

  • Gérard Cornuéjols

    (Tepper School of Business, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, and LIF, Faculté des Sciences de Luminy, 13288 Marseille, France)

  • Yanjun Li

    (Krannert School of Management, Purdue University, West Lafayette, Indiana 47907)

Abstract

Mixed-integer Gomory cuts have become an integral part of state-of-the-art software for solving mixed-integer linear programming problems. Therefore, improvements in the performance of these cutting planes can be of great practical value. In this paper, we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed-integer Gomory cuts. This is motivated by the fact that in a mixed-integer Gomory cut, the coefficient of an integer variable lies between 0 and 1, whereas for a continuous variable, there is no upper bound. The heuristic tries to reduce the coefficients of the continuous variables. We call the resulting cuts reduce-and-split cuts. We found that on several test problems, reduce-and-split cuts can substantially enhance the performance of a branch-and-bound algorithm.

Suggested Citation

  • Kent Andersen & Gérard Cornuéjols & Yanjun Li, 2005. "Reduce-and-Split Cuts: Improving the Performance of Mixed-Integer Gomory Cuts," Management Science, INFORMS, vol. 51(11), pages 1720-1732, November.
    • Handle: RePEc:inm:ormnsc:v:51:y:2005:i:11:p:1720-1732
    DOI: 10.1287/mnsc.1050.0382
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    References listed on IDEAS

    as
    1. Balas, Egon & Jeroslow, Robert G., 1980. "Strengthening cuts for mixed integer programs," European Journal of Operational Research, Elsevier, vol. 4(4), pages 224-234, April.
    2. Egon Balas, 1971. "Intersection Cuts—A New Type of Cutting Planes for Integer Programming," Operations Research, INFORMS, vol. 19(1), pages 19-39, February.
    3. Gérard Cornuéjols & Yanjun Li & Dieter Vandenbussche, 2003. "K-Cuts: A Variation of Gomory Mixed Integer Cuts from the LP Tableau," INFORMS Journal on Computing, INFORMS, vol. 15(4), pages 385-396, November.
    4. Harlan Crowder & Ellis L. Johnson & Manfred Padberg, 1983. "Solving Large-Scale Zero-One Linear Programming Problems," Operations Research, INFORMS, vol. 31(5), pages 803-834, October.
    5. Egon Balas & Sebastián Ceria & Gérard Cornuéjols, 1996. "Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework," Management Science, INFORMS, vol. 42(9), pages 1229-1246, September.
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    Cited by:

    1. Egon Balas & Gérard Cornuéjols & Tamás Kis & Giacomo Nannicini, 2013. "Combining Lift-and-Project and Reduce-and-Split," INFORMS Journal on Computing, INFORMS, vol. 25(3), pages 475-487, August.
    2. Konstantinos G. Zografos & Michael A. Madas & Konstantinos N. Androutsopoulos, 2017. "Increasing airport capacity utilisation through optimum slot scheduling: review of current developments and identification of future needs," Journal of Scheduling, Springer, vol. 20(1), pages 3-24, February.
    3. Santanu S. Dey & Andrea Lodi & Andrea Tramontani & Laurence A. Wolsey, 2014. "On the Practical Strength of Two-Row Tableau Cuts," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 222-237, May.
    4. John Heiney & Ryan Lovrien & Nicholas Mason & Irfan Ovacik & Evan Rash & Nandini Sarkar & Harry Travis & Zhenying Zhao & Kalani Ching & Shamin Shirodkar & Karl Kempf, 2021. "Intel Realizes $25 Billion by Applying Advanced Analytics from Product Architecture Design Through Supply Chain Planning," Interfaces, INFORMS, vol. 51(1), pages 9-25, February.
    5. Basu, Amitabh & Molinaro, Marco, 2015. "Characterization of the split closure via geometric lifting," European Journal of Operational Research, Elsevier, vol. 243(3), pages 745-751.
    6. Wesselmann, Franz & Koberstein, Achim & Suhl, Uwe H., 2011. "Pivot-and-reduce cuts: An approach for improving Gomory mixed-integer cuts," European Journal of Operational Research, Elsevier, vol. 214(1), pages 15-26, October.
    7. Haroldo G. Santos & Túlio A. M. Toffolo & Rafael A. M. Gomes & Sabir Ribas, 2016. "Integer programming techniques for the nurse rostering problem," Annals of Operations Research, Springer, vol. 239(1), pages 225-251, April.
    8. Santanu S. Dey & Jean-Philippe P. Richard, 2008. "Facets of Two-Dimensional Infinite Group Problems," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 140-166, February.

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