Skip to main content
SpringerLink
Log in

Cut generation through binarization

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

For a mixed integer linear program where all integer variables are bounded, we study a reformulation introduced by Roy that maps general integer variables to a collection of binary variables. We study theoretical properties and empirical strength of rank-2 simple split cuts of the reformulation. We show that for a pure integer problem with two integer variables, these cuts are sufficient to obtain the integer hull of the problem, but that this does not generalize to problems in higher dimensions. We also give an algorithm to compute an approximation of the rank-2 simple split cut closure. We report empirical results on 22 benchmark instances showing that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice-free cut closure. It gives a better bound than the split closure on 6 instances while it is weaker on 9 instances, for an average gap closed 3.8 % smaller than the one for the split closure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Oper. Res. Lett. 34, 361–372 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersen, K., Louveaux, Q., Weismantel, R., Wolsey L.: Cutting planes from two rows of a simplex tableau. In: Fischetti, M., Williamson, D.P., (eds.) Proceedings of IPCO XII, Lecture Notes in Computer Science 4513, pp. 1–15. (2007)

  3. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–323 (1993)

    Article  MATH  Google Scholar 

  4. Balas, E., Jeroslow, R.G.: Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4(4), 224–234 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. 113, 219–240 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: On the relative strength of split triangle and quadrilateral cuts. Math. Program. 126, 281–314 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bonami, P.: On optimizing over lift-and-project closures. Math. Program. Comput 4, 151–179 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bonami, P., Margot, F.: Cut Generation Through Binarization, Extended Abstract. In: J. Lee and J. Vygen (eds.) Proceedings of IPCO XVII, Lecture Notes in Computer Science (to appear)

  9. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: An updated mixed integer programming library: MIPLIB 3.0. Optima 58, 12–15 (1998)

    Google Scholar 

  10. Chen, B., Küçükyavuz, S., Sen, S.: Finite disjunctive programming characterizations for general mixed-integer linear programs. Oper. Res. 59, 202–210 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cornuéjols, G., Margot, F.: On the facets of mixed integer programs with two integer variables and two constraints. Math. Program. 120, 419–456 (2009)

    Article  Google Scholar 

  13. Dash, S., Günlük, O., Lodi, A.: MIR closures of polyhedral sets. Math. Program. 121, 33–60 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dash, S., Günlük, O., Molinaro, M.: On the relative strength of different generalizations of split cuts. Working paper (2012)

  15. Dash, S., Günlük, O., Vielma, J.P.: Computational experiments with cross and crooked cross cuts. Working paper (2011)

  16. Fischetti, M., Salvagnin, D.: Approximating the split closure. INFORMS J. Comput 25, 808–819 (2013)

    Article  MathSciNet  Google Scholar 

  17. Fischetti, M., Salvagnin, D.: Private communication

  18. Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)

    Google Scholar 

  19. Li, Y., Richard, J.-P.P.: Cook, Kannan and Schrijver’s example revisited. Discrete Optim. 5, 724–734 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Louveaux, Q., Poirrier, L., Salvagnin, D.: The strength of multi-row models. Working paper (2012)

  21. Owen, H., Mehrotra, S.: On the value of binary expansions for general mixed-integer linear programs. Oper. Res. 50, 810–819 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. http://wpweb2.tepper.cmu.edu/fmargot/OSCLIB/osc.htm

  23. Roy, J.-S.: “Binarize and Project” to generate cuts for general mixed-integer programs. Algorithmic Oper. Res. 2, 37–51 (2007)

Download references

Acknowledgments

F. Margot was supported by ONR Grant N00014-12-10032 and did part of this work while visiting the Ecole Polytechnique Fédérale de Lausanne (Switzerland).

Author information

Authors and Affiliations

  1. IBM ILOG CPLEX, Madrid, Spain

    Pierre Bonami

  2. LIF, Aix-Marseille Université, Marseille, France

    Pierre Bonami

  3. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    François Margot

Authors
  1. Pierre Bonami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. François Margot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Margot.

Additional information

Supported by ONR Grant N00014-12-10032.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonami, P., Margot, F. Cut generation through binarization. Math. Program. 154, 197–223 (2015). https://doi.org/10.1007/s10107-015-0924-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0924-z

Keywords

Mathematics Subject Classification

Search

Navigation