IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v54y2006i4p756-766.html
   My bibliography  Save this article

Combinatorial Benders' Cuts for Mixed-Integer Linear Programming

Author

Listed:
  • Gianni Codato

    (Department of Information Engineering, University of Padova, via Gradenigo 6/A, I-35100 Padova, Italy)

  • Matteo Fischetti

    (Department of Information Engineering, University of Padova, via Gradenigo 6/A, I-35100 Padova, Italy)

Abstract

Mixed-integer programs (MIPs) involving logical implications modeled through big-M coefficients are notoriously among the hardest to solve. In this paper, we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the big-M coefficients. Our solution scheme defines a master integer linear problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled” from the original MIP model. The master solutions are sent to a slave linear program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism closely resembles the Benders' one, but the cuts we produce are purely combinatorial and do not depend on the big-M values used in the MIP formulation. This produces an LP relaxation of the master problem which can be considerably tighter than the one associated with original MIP formulation. Computational results on two specific classes of hard-to-solve MIPs indicate that the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.

Suggested Citation

  • Gianni Codato & Matteo Fischetti, 2006. "Combinatorial Benders' Cuts for Mixed-Integer Linear Programming," Operations Research, INFORMS, vol. 54(4), pages 756-766, August.
    • Handle: RePEc:inm:oropre:v:54:y:2006:i:4:p:756-766
    DOI: 10.1287/opre.1060.0286
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1060.0286
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1060.0286?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Hak-Jin Kim & John Hooker, 2002. "Solving Fixed-Charge Network Flow Problems with a Hybrid Optimization and Constraint Programming Approach," Annals of Operations Research, Springer, vol. 115(1), pages 95-124, September.
    2. John Gleeson & Jennifer Ryan, 1990. "Identifying Minimally Infeasible Subsystems of Inequalities," INFORMS Journal on Computing, INFORMS, vol. 2(1), pages 61-63, February.
    3. Vipul Jain & Ignacio E. Grossmann, 2001. "Algorithms for Hybrid MILP/CP Models for a Class of Optimization Problems," INFORMS Journal on Computing, INFORMS, vol. 13(4), pages 258-276, November.
    4. Paul Rubin, 1997. "Solving mixed integer classification problems by decomposition," Annals of Operations Research, Springer, vol. 74(0), pages 51-64, November.
    5. Glover, Fred & Tangedahl, Lee, 1976. "Dynamic strategies for branch and bound," Omega, Elsevier, vol. 4(5), pages 571-576.
    6. John W. Chinneck, 2001. "Fast Heuristics for the Maximum Feasible Subsystem Problem," INFORMS Journal on Computing, INFORMS, vol. 13(3), pages 210-223, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jérémy Omer & Michael Poss, 2021. "Identifying relatively irreducible infeasible subsystems of linear inequalities," Annals of Operations Research, Springer, vol. 304(1), pages 361-379, September.
    2. Axel von Kamp & Steffen Klamt, 2014. "Enumeration of Smallest Intervention Strategies in Genome-Scale Metabolic Networks," PLOS Computational Biology, Public Library of Science, vol. 10(1), pages 1-13, January.
    3. Daniel Baena & Jordi Castro & Antonio Frangioni, 2020. "Stabilized Benders Methods for Large-Scale Combinatorial Optimization, with Application to Data Privacy," Management Science, INFORMS, vol. 66(7), pages 3051-3068, July.
    4. Timo Berthold & Jakob Witzig, 2021. "Conflict Analysis for MINLP," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 421-435, May.
    5. Dursun, Pınar & Taşkın, Z. Caner & Altınel, İ. Kuban, 2019. "The determination of optimal treatment plans for Volumetric Modulated Arc Therapy (VMAT)," European Journal of Operational Research, Elsevier, vol. 272(1), pages 372-388.
    6. René Brandenberg & Paul Stursberg, 2021. "Refined cut selection for benders decomposition: applied to network capacity expansion problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 383-412, December.
    7. Junhong Guo & William Pozehl & Amy Cohn, 2023. "A two-stage partial fixing approach for solving the residency block scheduling problem," Health Care Management Science, Springer, vol. 26(2), pages 363-393, June.
    8. Pedro Duarte Silva, A., 2017. "Optimization approaches to Supervised Classification," European Journal of Operational Research, Elsevier, vol. 261(2), pages 772-788.
    9. Christian Desrosiers & Philippe Galinier & Alain Hertz & Sandrine Paroz, 2009. "Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems," Journal of Combinatorial Optimization, Springer, vol. 18(2), pages 124-150, August.
    10. Adem, Jan & Gochet, Willy, 2006. "Mathematical programming based heuristics for improving LP-generated classifiers for the multiclass supervised classification problem," European Journal of Operational Research, Elsevier, vol. 168(1), pages 181-199, January.
    11. Gianpiero Canessa & Julian A. Gallego & Lewis Ntaimo & Bernardo K. Pagnoncelli, 2019. "An algorithm for binary linear chance-constrained problems using IIS," Computational Optimization and Applications, Springer, vol. 72(3), pages 589-608, April.
    12. Lihui Bai & Paul A. Rubin, 2009. "Combinatorial Benders Cuts for the Minimum Tollbooth Problem," Operations Research, INFORMS, vol. 57(6), pages 1510-1522, December.
    13. Kai Kellner & Marc E. Pfetsch & Thorsten Theobald, 2019. "Irreducible Infeasible Subsystems of Semidefinite Systems," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 727-742, June.
    14. Asparoukhov, Ognian K. & Krzanowski, Wojtek J., 2001. "A comparison of discriminant procedures for binary variables," Computational Statistics & Data Analysis, Elsevier, vol. 38(2), pages 139-160, December.
    15. Nascimento, Paulo Jorge & Silva, Cristóvão & Antunes, Carlos Henggeler & Moniz, Samuel, 2024. "Optimal decomposition approach for solving large nesting and scheduling problems of additive manufacturing systems," European Journal of Operational Research, Elsevier, vol. 317(1), pages 92-110.
    16. Saïd Hanafi & Fred Glover, 2002. "Resolution Search and Dynamic Branch-and-Bound," Journal of Combinatorial Optimization, Springer, vol. 6(4), pages 401-423, December.
    17. Brandner, Hubertus & Lessmann, Stefan & Voß, Stefan, 2013. "A memetic approach to construct transductive discrete support vector machines," European Journal of Operational Research, Elsevier, vol. 230(3), pages 581-595.
    18. Roberto Rossi & S. Armagan Tarim & Brahim Hnich & Steven Prestwich & Semra Karacaer, 2010. "Scheduling internal audit activities: a stochastic combinatorial optimization problem," Journal of Combinatorial Optimization, Springer, vol. 19(3), pages 325-346, April.
    19. Riise, Atle & Mannino, Carlo & Lamorgese, Leonardo, 2016. "Recursive logic-based Benders’ decomposition for multi-mode outpatient scheduling," European Journal of Operational Research, Elsevier, vol. 255(3), pages 719-728.
    20. Castro, Pedro M. & Oliveira, José F., 2011. "Scheduling inspired models for two-dimensional packing problems," European Journal of Operational Research, Elsevier, vol. 215(1), pages 45-56, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:54:y:2006:i:4:p:756-766. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.