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Journal of Global Optimization

, Volume 70, Issue 1, pp 289–306 | Cite as

Generalized coefficient strengthening cuts for mixed integer programming

  • Wei-Kun Chen
  • Liang Chen
  • Mu-Ming Yang
  • Yu-Hong Dai
Article
  • 209 Downloads

Abstract

Cutting plane methods are an important component in solving the mixed integer programming (MIP). By carefully studying the coefficient strengthening method, which is originally a presolving method, we are able to generalize this method to generate a family of valid inequalities called generalized coefficient strengthening (GCS) inequalities. The invariant property of the GCS inequalities is established under bound substitutions. Furthermore, we develop a separation algorithm for finding the violated GCS inequalities for a general mixed integer set. The separation algorithm is proved to have the polynomial time complexity. Extensive numerical experiments are made on standard MIP test sets, which demonstrate the usefulness of the resulting GCS separator.

Keywords

Mixed integer programming Cutting plane method Separation algorithm Coefficient strengthening 

References

  1. 1.
    Gomory, R.: An Algorithm for the Mixed Integer Problem. RM-2597. The Rand Corporation, Santa Monica (1960)Google Scholar
  2. 2.
    Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Letchford, A.N., Lodi, A.: Strengthening Chvátal–Gomory cuts and Gomory fractional cuts. Oper. Res. Lett. 30, 74–82 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Koster, A.M.C.A., Zymolka, A., Kutschka, M.: Algorithms to separate \(\{0,\frac{1}{2}\}\)-Chvátal–Gomory cuts. Algorithmica 55, 375–391 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hoffman, K.L., Padberg, M.: Improving LP-representations of zero-one linear programs for branch-and-cut. ORSA J. Comput. 3, 121–134 (1991)CrossRefzbMATHGoogle Scholar
  6. 6.
    Crowder, H., Johnson, E.L., Padberg, M.: Solving large-scale zero-one linear programming problems. Oper. Res. 31, 803–834 (1983)CrossRefzbMATHGoogle Scholar
  7. 7.
    Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.P.: Conflict graphs in solving integer programming problems. Eur. J. Oper. Res. 121, 40–55 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Balas, E.: Disjunctive programming. Ann. Discret. Math. 5, 3–51 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Weismantel, R.: On the 0/1 knapsack polytope. Math. Program. 77, 49–68 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Van Roy, T.J., Wolsey, L.A.: Valid inequalities and separation for uncapacitated fixed charge networks. Oper. Res. Lett. 4, 105–112 (1985)CrossRefzbMATHGoogle Scholar
  11. 11.
    Padberg, M.W., Van Roy, T.J., Wolsey, L.A.: Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted flow cover inequalities for mixed 0–1 integer programs. Math. Program. 85, 439–467 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted cover inequalities for 0–1 integer programs: computation. INFORMS J. Comput. 10, 427–437 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Achterberg, T., Raack, C.: The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs. Math. Program. Comput. 2, 125–165 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Achterberg, T., Wunderling, R.: Mixed integer programming: analyzing 12 years of progress. In: Jnger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 449–481. Springer, Berlin (2013)CrossRefGoogle Scholar
  16. 16.
  17. 17.
    Achterberg, T., Bixby, R.E., Gu, Z., Rothberg, E., Weninger, D.: Presolve reductions in mixed integer programming. ZIB Report 16–44, Zuse Institute Berlin, (2016)Google Scholar
  18. 18.
    Savelsbergh, M.W.P.: Preprocessing and probing techniques for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Achterberg, T.: Constraint Integer Programming. Ph.D. thesis, Technische Universität, Berlin (2007). https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/1112
  20. 20.
    Atamtürk, A., Rajan, D.: On splittable and unsplittable flow capacitated network design arc-set polyhedra. Math. Program. 92, 315–333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Magnanti, T.L., Mirchandani, P., Vachani, R.: The convex hull of two core capacitated network design problems. Math. Program. 60, 233–250 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gamrath, G., Fischer, T., Gally, T., Gleixner, A.M., Hendel, G., Koch, T., Maher, S.J., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C.: The SCIP optimization suite 3.2. ZIB Report 15–60, Zuse Institute Berlin, (2016)Google Scholar
  23. 23.
    Bixby, R., Ceria, S., McZeal, C., Savelsbergh, M.: An updated mixed integer programming library: MIPLIB 3.0. Optima 54, 12–15 (1998)Google Scholar
  24. 24.
    Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Oper. Res. Lett. 34, 361–372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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
  26. 26.
    Wunderling, R.: Paralleler und objektorientierter simplex. Ph.D. thesis, Technische Universität Berlin, (1996). https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/538
  27. 27.
    Wolter, K.: Implementation of cutting plane separators for mixed integer programs. Master’s thesis, Technische Universität Berlin, (2006)Google Scholar
  28. 28.
    Beate, B., Günlük, O., Wolsey, L.A.: Designing private line networks: polyhedral analysis and computation. CORE Discussion Paper 9647, Université Catholique de Louvain, (1996)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Scientific/Engineering Computing, State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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