Learning a Classification of Mixed-Integer Quadratic Programming Problems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10848)


Within state-of-the-art solvers such as IBM-CPLEX, the ability to solve both convex and nonconvex Mixed-Integer Quadratic Programming (MIQP) problems to proven optimality goes back few years, yet presents unclear aspects. We are interested in understanding whether for solving an MIQP it is favorable to linearize its quadratic part or not. Our approach exploits machine learning techniques to learn a classifier that predicts, for a given instance, the most suitable resolution method within CPLEX’s framework. We aim as well at gaining first methodological insights about the instances’ features leading this discrimination. We examine a new dataset and discuss different scenarios to integrate learning and optimization. By defining novel measures, we interpret and evaluate learning results from the optimization point of view.


  1. 1.
    Kruber, M., Lübbecke, M.E., Parmentier, A.: Learning when to use a decomposition. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 202–210. Springer, Cham (2017). Scholar
  2. 2.
    Khalil, E.B., Dilkina, B., Nemhauser, G., Ahmed, S., Shao, Y.: Learning to run heuristics in tree search. In: 26th International Joint Conference on Artificial Intelligence (IJCAI) (2017)Google Scholar
  3. 3.
    Lodi, A., Zarpellon, G.: On learning and branching: a survey. TOP 25(2), 207–236 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hutter, F., Hoos, H.H., Leyton-Brown, K.: Automated configuration of mixed integer programming solvers. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 186–202. Springer, Heidelberg (2010). Scholar
  5. 5.
    Hutter, F., Xu, L., Hoos, H.H., Leyton-Brown, K.: Algorithm runtime prediction: methods and evaluation. Artif. Intell. 206, 79–111 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74(2), 121–140 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonami, P., Lejeune, M.A.: An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Oper. Res. 57(3), 650–670 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
  9. 9.
    Bliek, C., Bonami, P., Lodi, A.: Solving mixed-integer quadratic programming problems with IBM-CPLEX: a progress report. In: Proceedings of the Twenty-Sixth RAMP Symposium, pp. 16–17 (2014)Google Scholar
  10. 10.
    Fourer, R.: Quadratic optimization mysteries, part 1: two versions (2015).
  11. 11.
    Gupta, O.K., Ravindran, A.: Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31(12), 1533–1546 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., et al.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5(2), 186–204 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Land, A., Doig, A.: An automatic method of solving discrete programming problems. Econometrica 28, 497–520 (1960)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36(3), 307–339 (1986)MathSciNetCrossRefGoogle Scholar
  15. 15.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefGoogle Scholar
  16. 16.
    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numerica 22, 1–131 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    MATLAB: Version 9.1.0 (2016). The MathWorks Inc., Natick, MassachusettsGoogle Scholar
  18. 18.
    Puchinger, J., Raidl, G.R., Pferschy, U.: The multidimensional Knapsack problem: structure and algorithms. INFORMS J. Comput. 22(2), 250–265 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lodi, A., Tramontani, A.: Performance variability in mixed-integer programming. In: Theory Driven by Influential Applications, INFORMS, pp. 1–12 (2013)Google Scholar
  20. 20.
    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)zbMATHGoogle Scholar
  22. 22.
    Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001)CrossRefGoogle Scholar
  23. 23.
    Geurts, P., Ernst, D., Wehenkel, L.: Extremely randomized trees. Mach. Learn. 63(1), 3–42 (2006)CrossRefGoogle Scholar
  24. 24.
    Friedman, J.H.: Stochastic gradient boosting. Comput. Stat. Data Anal. 38(4), 367–378 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Furini, F., Traversi, E., Belotti, P., Frangioni, A., Gleixner, A., Gould, N., Liberti, L., Lodi, A., Misener, R., Mittelmann, H., Sahinidis, N., Vigerske, S., Wiegele, A.: QPLIB: a library of quadratic programming instances. Technical report (2017). Available at Optimization OnlineGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CPLEX Optimization, IBM SpainMadridSpain
  2. 2.CERCÉcole Polytechnique MontréalMontrealCanada

Personalised recommendations