Abstract

Branch-and-bound methods for mixed-integer programming (MIP) are traditionally based on solving a linear programming (LP) relaxation and branching on a variable which takes a fractional value in the (single) computed relaxation optimum. In this paper we study branching strategies for mixed-integer programs that exploit the knowledge of multiple alternative optimal solutions (a cloud) of the current LP relaxation. These strategies naturally extend state-of-the-art methods like strong branching, pseudocost branching, and their hybrids.

We show that by exploiting dual degeneracy, and thus multiple alternative optimal solutions, it is possible to enhance traditional methods. We present preliminary computational results, applying the newly proposed strategy to full strong branching, which is known to be the MIP branching rule leading to the fewest number of search nodes. It turns out that cloud branching can reduce the mean running time by up to 30% on standard test sets.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benichou, M., Gauthier, J., Girodet, P., Hentges, G., Ribiere, G., Vincent, O.: Experiments in mixed-integer programming. Mathematical Programming 1, 76–94 (1971)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Linderoth, J.T., Savelsbergh, M.W.P.: A computational study of strategies for mixed integer programming. INFORMS Journal on Computing 11, 173–187 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Operations Research Letters 33, 42–54 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Achterberg, T.: Constraint Integer Programming. PhD thesis, Technische Universität Berlin (2007), http://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/1018
  5. 5.
    Bixby, R., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: MIP: Theory and practice – closing the gap. In: Powell, M., Scholtes, S. (eds.) Systems Modelling and Optimization: Methods, Theory, and Applications, pp. 19–49. Kluwer Academic Publisher (2000)Google Scholar
  6. 6.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: Finding cuts in the TSP (A preliminary report). Technical Report 95-05, DIMACS (1995)Google Scholar
  7. 7.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, USA (2007)Google Scholar
  8. 8.
    Fischetti, M., Monaci, M.: Branching on nonchimerical fractionalities. OR Letters 40(3), 159–164 (2012)MathSciNetMATHGoogle Scholar
  9. 9.
    Achterberg, T., Berthold, T.: Hybrid branching. In: van Hoeve, W.-J., Hooker, J.N. (eds.) CPAIOR 2009. LNCS, vol. 5547, pp. 309–311. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Patel, J., Chinneck, J.W.: Active-constraint variable ordering for faster feasibility of mixed integer linear programs. Mathematical Programming 110, 445–474 (2007)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Karamanov, M., Cornuéjols, G.: Branching on general disjunctions. Mathematical Programming 128(1-2), 403–436 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Li, C.M., Anbulagan: Look-ahead versus look-back for satisfiability problems. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 341–355. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. 13.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of the 38th Annual Design Automation Conference (DAC 2001), pp. 530–535 (2001), doi:10.1145/378239.379017Google Scholar
  14. 14.
    Kılınç Karzan, F., Nemhauser, G.L., Savelsbergh, M.W.P.: Information-based branching schemes for binary linear mixed-integer programs. Mathematical Programming Computation 1(4), 249–293 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fischetti, M., Monaci, M.: Backdoor branching. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 183–191. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    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. Mathematical Programming Computation 3, 103–163 (2011), http://miplib.zib.de MathSciNetCrossRefGoogle Scholar
  17. 17.
    Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Operations Research Letters 34(4), 1–12 (2006), http://miplib.zib.de/miplib2003/ MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. Journal of Global Optimization 14, 217–249 (1999), doi:10.1023/A:1008312714792MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Caprara, A., Locatelli, M.: Global optimization problems and domain reduction strategies. Mathematical Programming 125, 123–137 (2010), doi:10.1007/s10107-008-0263-4MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Fischetti, M., Glover, F., Lodi, A.: The feasibility pump. Mathematical Programming 104(1), 91–104 (2005), doi:10.1007/s10107-004-0570-3MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Achterberg, T.: LP basis selection and cutting planes. Presentation Slides from MIP 2010 Conference in Atlanta (2010), http://www2.isye.gatech.edu/mip2010/program/program.pdf
  22. 22.
    Achterberg, T.: SCIP: Solving Constraint Integer Programs. Mathematical Programming Computation 1(1), 1–41 (2009), doi:10.1007/s12532-008-0001-1MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Wunderling, R.: Paralleler und objektorientierter Simplex-Algorithmus. PhD thesis, Technische Universität Berlin (1996)Google Scholar
  24. 24.
  25. 25.
    Czyzyk, J., Mesnier, M., Moré, J.: The NEOS server. IEEE Computational Science & Engineering 5(3), 68–75 (1998), http://www.neos-server.org/neos/ CrossRefGoogle Scholar
  26. 26.
    Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.: An updated mixed integer programming library: MIPLIB 3.0. Optima (58), 12–15 (1998), http://miplib.zib.de/miplib3/miplib.html
  27. 27.
    Cohen, P.R.: Empirical Methods for Artificial Intelligence. MIT Press (1995)Google Scholar
  28. 28.
    Gamrath, G.: Improving strong branching by propagation. In: Gomes, C., Sellmann, M. (eds.) CPAIOR 2013. LNCS, vol. 7874, pp. 347–354. Springer, Heidelberg (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Timo Berthold
    • 1
  • Domenico Salvagnin
    • 2
  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.DEIPadovaItaly

Personalised recommendations