Advertisement

Abstract

Current state-of-the-art MIP technology lacks a powerful modeling language based on global constraints, a tool which has long been standard in constraint programming. In general, even basic semantic information about variables and constraints is hidden from the underlying solver. For example, in a network design model with unsplittable flows, both routing and arc capacity variables could be binary, and the solver would not be able to distinguish between the two semantically different groups of variables by looking at type alone. If available, such semantic partitioning could be used by different parts of the solver, heuristics in primis, to improve overall performance. In the present paper we will describe several heuristic procedures, all based on the concept of partition refinement, to automatically recover semantic variable (and constraint) groups from a flat MIP model. Computational experiments on a heterogeneous testbed of models, whose original higher-level partition is known a priori, show that one of the proposed methods is quite effective.

References

  1. 1.
    Achterberg, T.: Constraint integer programming. Ph.D. thesis, Technische Universität Berlin (2007)Google Scholar
  2. 2.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Achterberg, T., Raack, C.: The MCF-separator: detecting and exploiting multi-commodity flow structures in MIPs. Math. Program. Comput. 2(2), 125–165 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aho, A., Hopcroft, J., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  5. 5.
    Althaus, E., Bockmayr, A., Elf, M., Jünger, M., Kasper, T., Mehlhorn, K.: SCIL - symbolic constraints in integer linear programming. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, p. 75. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Audet, C., Brimberg, J., Hansen, P., Digabel, S.L., Mladenovic, N.: Pooling problem: alternate formulations and solution methods. Manag. Sci. 50(6), 761–776 (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Avella, P., Boccia, M.: A cutting plane algorithm for the capacitated facility location problem. Comput. Optim. Appl. 43(1), 39–65 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baker, K.R., Trietsch, D.: Principles of Sequencing and Scheduling. Wiley, New York (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bley, A., Boland, N., Fricke, C., Froyland, G.: A strengthened formulation and cutting planes for the open pit mine production scheduling problem. Comput. Oper. Res. 37, 1641–1647 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Borndörfer, R., Liebchen, C.: When periodic timetables are suboptimal. In: Kalcsics, J., Nickel, S. (eds.) Operations Research Proceedings 2007, pp. 449–454. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Brooks, J.P.: Support vector machines with the ramp loss and the hard margin loss. Oper. Res. 59(2), 467–479 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cire, A., Hooker, J.N., Yunes, T.: Modeling with metaconstraints and semantic typing of variables. INFORMS J. Comput. (to appear)Google Scholar
  13. 13.
    Darga, P.T., Liffiton, M.H., Sakallah, K.A., Markov, I.L.: Exploiting structure in symmetry detection for CNF. In: Proceedings of the 41th Design Automation Conference, DAC 2004, San Diego, CA, USA, 7–11 June 2004, pp. 530–534 (2004)Google Scholar
  14. 14.
    Darga, P.T., Sakallah, K.A., Markov, I.L.: Faster symmetry discovery using sparsity of symmetries. In: Proceedings of the 45th Design Automation Conference, DAC 2008, Anaheim, CA, USA, 8–13 June 2008, pp. 149–154 (2008)Google Scholar
  15. 15.
    Fischetti, M., Monaci, M., Salvagnin, D.: Mixed-integer linear programming heuristics for the prepack optimization problem. Discrete Optimization (to appear)Google Scholar
  16. 16.
    Fischetti, M., Salvagnin, D.: Feasibility pump 2.0. Math. Program. Comput. 1(2–3), 201–222 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Thomson, Stamford (2003)zbMATHGoogle Scholar
  18. 18.
    Gamrath, G., Berthold, T., Heinz, S., Winkler, M.: Structure-based primal heuristics for mixed integer programming. In: Fujisawa, K., Shinano, Y., Waki, H. (eds.) Optimization in the Real World. Mathematics for Industry, vol. 13, pp. 37–53. Springer Japan, Tokyo (2016)CrossRefGoogle Scholar
  19. 19.
    GUROBI: GUROBI 6.0 User’s Manual (2015)Google Scholar
  20. 20.
    Hooker, J.N.: Integrated Methods for Optimization. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  21. 21.
    Hooker, J.N.: Logic-based modeling. In: Appa, G., Pitsoulis, M., Leonidas, S., Williams, H.P. (eds.) Handbook on Modelling for Discrete Optimization, pp. 61–102. Springer, US (2006)Google Scholar
  22. 22.
    Hooker, J.N.: Hybrid modeling. In: van Hentenryck, P., Milano, M. (eds.) Hybrid Optimization: the Ten Years of CPAIOR, pp. 11–62. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  23. 23.
    Hoskins, M., Masson, R., Gauthier Melançon, G., Mendoza, J.E., Meyer, C., Rousseau, L.-M.: The prepack optimization problem. In: Simonis, H. (ed.) CPAIOR 2014. LNCS, vol. 8451, pp. 136–143. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  24. 24.
    Hüffner, F., Betzler, N., Niedermeier, R.: Separator-based data reduction for signed graph balancing. J. Comb. Optim. 20, 335–360 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    IBM ILOG: CPLEX 12.6.2 User’s Manual (2015)Google Scholar
  26. 26.
    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 - mixed integer programming library version 5. Math. Program. Comput. 3, 103–163 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lodi, A.: Mixed integer programming computation. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008 - From the Early Years to the State-of-the-Art, pp. 619–645. Springer, Heidelberg (2010)Google Scholar
  28. 28.
    Margot, F.: Symmetry in integer linear programming. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming. Springer, Heidelberg (2009)Google Scholar
  29. 29.
    McKay, B.D.: Practical Graph Isomorphism (1981)Google Scholar
  30. 30.
    Milano, M.: Constraint and Integer Programming: Toward a Unified Methodology. Kluwer Academic Publishers, Norwell (2003)zbMATHGoogle Scholar
  31. 31.
    Mitra, G., Lucas, C., Moody, S., Hadjiconstantinou, E.: Tools for reformulating logical forms into zero-one mixed integer programs. Eur. J. Oper. Res. 72(2), 262–276 (1994)CrossRefzbMATHGoogle Scholar
  32. 32.
    Rossi, F., van Beek, P., Walsh, T. (eds.): Handbook of Constraint Programming. Foundations of Artificial Intelligence. Elsevier, Amsterdam (2006)Google Scholar
  33. 33.
    Salvagnin, D.: A dominance procedure for Integer programming. Master’s thesis, University of Padova (2005)Google Scholar
  34. 34.
    Salvagnin, D.: Detecting and exploiting permutation structures in MIPs. In: Simonis, H. (ed.) CPAIOR 2014. LNCS, vol. 8451, pp. 29–44. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  35. 35.
    Salvagnin, D., Walsh, T.: A hybrid MIP/CP approach for multi-activity shift scheduling. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 633–646. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  36. 36.
    Savelsbergh, M.W.P.: Preprocessing and probing for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Stuckey, P.J., Tack, G.: MiniZinc with functions. In: Gomes, C., Sellmann, M. (eds.) CPAIOR 2013. LNCS, vol. 7874, pp. 268–283. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  38. 38.
    Sun, M., Aronson, J.E., McKeown, P.G., Drinka, D.A.: A tabu search heuristic procedure for the fixed charge transportation problem. Eur. J. Oper. Res. 106, 441–456 (1998)CrossRefzbMATHGoogle Scholar
  39. 39.
    Yunes, T.: CuSPLIB 1.0: A library of single-machine cumulative scheduling problems (2009). http://moya.bus.miami.edu/tallys/cusplib/
  40. 40.
    Yunes, T., Aron, I.D., Hooker, J.N.: An integrated solver for optimization problems. Oper. Res. 58(2), 342–356 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    SNDlib (2006). http://sndlib.zib.de

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.IBM ItalySegrateItaly
  2. 2.DEIUniversity of PadovaPaduaItaly

Personalised recommendations