Computational Optimization and Applications

, Volume 65, Issue 3, pp 545–566 | Cite as

On handling indicator constraints in mixed integer programming

  • Pietro Belotti
  • Pierre Bonami
  • Matteo Fischetti
  • Andrea Lodi
  • Michele Monaci
  • Amaya Nogales-Gómez
  • Domenico Salvagnin
Article

Abstract

Mixed integer programming (MIP) is commonly used to model indicator constraints, i.e., constraints that either hold or are relaxed depending on the value of a binary variable. Unfortunately, those models tend to lead to weak continuous relaxations and turn out to be unsolvable in practice; this is what happens, for e.g., in the case of Classification problems with Ramp Loss functions that represent an important application in this context. In this paper we show the computational evidence that a relevant class of these Classification instances can be solved far more efficiently if a nonlinear, nonconvex reformulation of the indicator constraints is used instead of the linear one. Inspired by this empirical and surprising observation, we show that aggressive bound tightening is the crucial ingredient for solving this class of instances, and we devise a pair of computationally effective algorithmic approaches that exploit it within MIP. One of these methods is currently part of the arsenal of IBM-Cplex  since version 12.6.1. More generally, we argue that aggressive bound tightening is often overlooked in MIP, while it represents a significant building block for enhancing MIP technology when indicator constraints and disjunctive terms are present.

Keywords

Mixed-integer linear programming Mixed-integer quadratic programming Indicator constraints 

Notes

Acknowledgments

The research of Fischetti, Monaci and Salvagnin was supported by the University of Padova (Progetto di Ateneo “Exploiting randomness in Mixed Integer Linear Programming”). The research of Fischetti, Lodi, Monaci and Salvagnin was supported by MiUR, Italy (PRIN Project “Mixed-Integer Nonlinear Optimization: Approaches and Applications”). The work of Nogales-Gómez was supported by an STSM Grant from COST Action TD1207. The authors are grateful to Andrea Tramontani and Sven Wiese for many interesting discussions on the subject and to two anonymous referees for a careful reading and many useful comments that helped improving the paper.

References

  1. 1.
    Andersen, E., Andersen, K.: Presolving in linear programming. Math. Program. 71, 221–245 (1995)MathSciNetMATHGoogle Scholar
  2. 2.
    Balas, E.: Disjunctive programming. Ann. Discret. Math. 5, 3–51 (1979)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5, 186–2004 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bonami, P., Kilinc, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) Hot Topics in Mixed Integer Nonlinear Programming, IMA Volumes, pp. 1–40. Springer, Berlin (2012)CrossRefGoogle Scholar
  6. 6.
    Bonami, P., Lodi, A., Tramontani, A., Wiese, S.: On mathematical programming with indicator constraints. Math. Program. 151, 191–223 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
  8. 8.
    Brooks, J.P.: Support vector machines with the ramp loss and the hard margin loss. Oper. Res. 59(2), 467–479 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Carrizosa, E., Romero Morales, D.: Supervised classification and mathematical optimization. Comput. Oper. Res. 40, 150–165 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
  11. 11.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595–614 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Collobert, R., Sinz, F., Weston, J., Bottou, L.: Trading convexity for scalability. In: Proceedings of the 23rd International Conference on Machine Learning, pp. 201–208 (2006)Google Scholar
  13. 13.
    Couenne, v. branch/CouenneClassifier, r1046. https://projects.coin-or.org/Couenne
  14. 14.
    Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, Cambridge (2000)CrossRefMATHGoogle Scholar
  15. 15.
    Davis, E.: Constraint propagation with interval labels. Artif. Intell. 32(3), 281–331 (1987)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Duran, M.A., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    FICO Xpress Optimization Suite, v. 7.8. http://www.fico.com/xpress
  18. 18.
    Fischetti, M., Monaci, M.: Exploiting erraticism in search. Oper. Res. 62, 114–122 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Grossmann, I.E., Trespalacios, F.: Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming. AIChE J. 59(9), 3276–3295 (2013)CrossRefGoogle Scholar
  20. 20.
    Gurobi, v. 6.0.2. http://www.gurobi.com
  21. 21.
  22. 22.
  23. 23.
    Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: Miplib 2010. Math. Program. Comput. 3, 103–163 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lodi, A., Tramontani, A.: Performance variability in mixed-integer programming. In: Topaloglu, H. (ed.) TutORials in Operations Research: Theory Driven by Influential Applications, pp. 1–12. INFORMS, Catonsville (2013)Google Scholar
  25. 25.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Messine, F.: Deterministic global optimization using interval constraint propagation techniques. RAIRO-RO 38(4), 277–294 (2004)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Quesada, I., Grossmann, I.E.: An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)CrossRefGoogle Scholar
  28. 28.
    Savelsbergh, M.W.P.: Preprocessing and probing techniques for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Shen, X., Tseng, G.C., Zhang, X., Wong, W.H.: On \(\psi \)-learning. J. Am. Stat. Assoc. 98, 724–734 (2003)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Boston (2002)CrossRefMATHGoogle Scholar
  31. 31.
    Wu, X., Kumar, V., Ross Quinlan, J., Ghosh, J., Yang, Q., Motoda, H., McLachlan, G.J., Ng, A., Liu, B., Yu, P.S., Zhou, Z.-H., Steinbach, M., Hand, D.J., Steinberg, D.: Top 10 algorithms in data mining. Knowl. Inf. Syst. 14, 1–37 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Pietro Belotti
    • 1
  • Pierre Bonami
    • 2
  • Matteo Fischetti
    • 3
  • Andrea Lodi
    • 4
    • 5
  • Michele Monaci
    • 4
  • Amaya Nogales-Gómez
    • 6
  • Domenico Salvagnin
    • 3
    • 7
  1. 1.FICOBirminghamUK
  2. 2.IBMMadridSpain
  3. 3.University of PadovaPaduaItaly
  4. 4.University of BolognaBolognaItaly
  5. 5.École Polytechnique de MontréalMontrealCanada
  6. 6.Mathematical and Algorithmic Sciences Lab, Huawei France R&DParisFrance
  7. 7.IBMMilanoItaly

Personalised recommendations