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Mathematical Programming

, Volume 113, Issue 2, pp 241–257 | Cite as

Projected Chvátal–Gomory cuts for mixed integer linear programs

  • Pierre Bonami
  • Gérard Cornuéjols
  • Sanjeeb Dash
  • Matteo Fischetti
  • Andrea Lodi
FULL LENGTH PAPER

Abstract

Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure integer linear program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal–Gomory (CG) separation problem as a mixed integer linear program (MILP) which is then solved by a general- purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory mixed integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal–Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, which can typically be solved in a reasonable amount of computing time.

Keywords

Mixed integer linear program Chvátal–Gomory cut Separation problem Projected polyhedron 

Mathematics Subject Classification (2000)

90C10 90C11 90C57 

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References

  1. 1.
    Ascheuer, N.: Hamiltonian path problems in the on-line optimization of flexible manufacturing systems. PhD Thesis, Technische Universität Berlin, Berlin (1995)Google Scholar
  2. 2.
    Ascheuer, N., Fischetti, M., Grötschel, M.: A polyhedral study of the asymmetric travelling salesman problem with time windows. Networks 36, 69–79 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Program. 58, 295–324 (1993)CrossRefGoogle Scholar
  4. 4.
    Balas, E., Saxena, A.: Optimizing over the Split Closure: Modeling and Theoretical Analysis, IMA “Hot Topics” Workshop: Mixed Integer Programming, Minneapolis, 25–29 July 2005Google Scholar
  5. 5.
    Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: MIPLIB 3.0, http://www.caam. rice.edu/~bixby/miplib/miplib.htmlGoogle Scholar
  6. 6.
    Bonami, P., Minoux,M.: Using rank-1 lift-and-project closures to generate cuts for 0-1MIPs, a computational investigation. Discrete Optim. 2, 288–307 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Caprara, A., Letchford, A.N.: On the separation of split cuts and related inequalities. Math. Program. 94, 279–294 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems C73. Discrete Math. 4, 305–337 (1973)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Codato, G., Fischetti, M.: Combinatorial Benders’ cuts for mixed-integer linear programming. Oper. Res. 54, 756–766 (2006)CrossRefMathSciNetGoogle Scholar
  10. 10.
    COIN-OR. www.coin-or.orgGoogle Scholar
  11. 11.
    Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cornuéjols, G., Li, Y.: On the rank of mixed 0,1 polyhedra.Math. Program. 91, 391–397 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cornuéjols, G., Li, Y.: A connection between cutting plane theory and the geometry of numbers. Math. Program. 93, 123–127 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dash, S., Günlük, O., Lodi, A.: On the MIR closure of polyhedra. IBM, T.J. Watson Research, Working paper (2005)Google Scholar
  15. 15.
    Eisenbrand, F.:On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fischetti, M., Lodi, A. : Optimizing over the first Ch closure. In: Jünger, M., Kaibel, V. (eds.) Integer Programming and Combinatorial Optimization—IPCO 2005, LNCS 3509., pp. 12–22. Springer, Berlin Heidelberg New York (2005)Google Scholar
  17. 17.
    Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull AMS 64, 275–278 (1958)MATHMathSciNetGoogle Scholar
  18. 18.
    Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming., pp. 269–302. McGraw-Hill, New York (1963)Google Scholar
  19. 19.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin Heidelberg New York (1988)MATHGoogle Scholar
  20. 20.
    ILOG Cplex 9.0: User’s Manual and Reference Manual, ILOG, S.A., http://www.ilog.com/ (2005)Google Scholar
  21. 21.
    Klau,G.W., Mützel,P.: Optimal labelling of point features in rectangular labellingmodels.Math. Program. 94, 435–458 (2003)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)MATHGoogle Scholar
  24. 24.
    Nemhauser,G.L.,Wolsey, L.A.:Arecursive procedure to generate all cuts for 0-1 mixed integer programs. Math. Program. 46, 379–390 (1990)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Schrijver, A.: On cutting planes. Ann. Discrete Math. 9, 291–296 (1980)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Pierre Bonami
    • 1
  • Gérard Cornuéjols
    • 2
    • 3
  • Sanjeeb Dash
    • 1
  • Matteo Fischetti
    • 4
  • Andrea Lodi
    • 5
  1. 1.IBM T.J. Watson Research CenterYorktown HeightsUSA
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  3. 3.LIFFaculté des Sciences de LuminyMarseilleFrance
  4. 4.DEIUniversity of PadovaPadovaItaly
  5. 5.DEISUniversity of BolognaBolognaItaly

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