Skip to main content
SpringerLink
Log in

Local cuts for mixed-integer programming

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

A general framework for cutting-plane generation was proposed by Applegate et al. in the context of the traveling salesman problem. The process considers the image of a problem space under a linear mapping, chosen so that a relaxation of the mapped problem can be solved efficiently. Optimization in the mapped space can be used to find a separating hyperplane, if one exists, and via substitution this gives a cutting plane in the original space. We extend this procedure to general mixed-integer programming problems, obtaining a range of possibilities for new sources of cutting planes. Some of these possibilities are explored computationally, both in floating-point arithmetic and in rational arithmetic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. It is easy to see that at least one of them must be violated by \(x^{*}\) if the original \(ax\le b\) was violated by \(x^{*}\).

References

  1. Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1, 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Oper. Res. Lett. 34, 1–12 (2006)

    Article  MathSciNet  Google Scholar 

  4. Althaus, E., Polzin, T., Daneshmand, S.V.: Improving linear programming approaches for the Steiner tree problem. In: Jansen, K., Margraf, M., Mastrolilli, M., Rolim, J.D.P. (eds.) Experimental and Efficient Algorithms, Second International Workshop, WEA 2003, pp. 1–14. Springer, Berlin (2003)

    Google Scholar 

  5. Applegate, D., Bixby, R.E., Chvátal, V., Cook, W.: TSP cuts which do not conform to the template paradigm. In: Computational Combinatorial Optimization, Optimal or Provably Near-Optimal Solutions [based on a Spring School], pp. 261–304. Springer, London (2001)

  6. Applegate, D., Bixby, R.E., Chvátal, V., Cook, W.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  7. Applegate, D., Cook, W., Dash, S., Espinoza, D.: Exact solutions to linear programming problems. Oper. Res. Lett. 35, 693–699 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Atamtürk, A.: Sequence independent lifting for mixed-integer programming. Oper. Res. 52, 487–490 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balas, E.: Disjunctive programming: Properties of the convex hull of feasible points. Discr. Appl. Math. 89, 3–44 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)

    Article  MATH  Google Scholar 

  11. Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.: Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. 113, 219–240 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bixby, R.E., Boyd, E.A., Indovina, R.R.: MIPLIB: a test set of mixed integer programming problems. SIAM News 25, 16 (1992)

    Google Scholar 

  14. Bixby, R.E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: Mixed-integer programming: a progress report. In: Grötschel, M. (ed.) The Sharpest Cut: The Impact of Manfred Padberg and His Work, pp. 309–325. SIAM, Philadelphia (2004)

    Chapter  Google Scholar 

  15. Boyd, E.A.: Generating Fenchel cutting planes for knapsack polyhedra. SIAM J. Optim. 3, 734–750 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boyd, E.A.: Fenchel cutting planes for integer programs. Oper. Res. 42, 53–64 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Buchheim, C., Liers, F., Oswald, M.: Local cuts revisited. Oper. Res. Lett. 36, 430–433 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Buchheim, C., Liers, F., Oswald, M.: Speeding up IP-based algorithms for constrained quadratic 0–1 optimization. Tech. Rep. zaik2008-578, Zentrum für Angewandte Informatik Köln, Germany (2008)

  19. Caprara, A., Fischetti, M.: 0,1/2-Chvátal-Gomory cuts. Math. Program. 74, 221–235 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Cook, W., Dash, S., Fukasawa, R., Goycoolea, M.: Numerically safe gomory mixed-integer cuts. INFORMS J. Comput. 21, 641–649 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cook, W., Koch, T., Steffy, D.E., Wolter, K.: An exact rational mixed-integer programming solver. In: Integer Programming and Combinatorial Optimization, 15th International IPCO Conference, IBM T. J. Watson Research Center, Yorktown Heights, New York, NY, USA, Lecture Notes in Computer Science, pp. 104–116. Springer, Berlin (2011)

  22. Cornuéjols, G., Li, Y., Vandenbussche, D.: K-cuts: a variation of Gomory mixed integer cuts from the LP tableau. INFORMS J. Comput. 15, 385–396 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Crowder, H.P., Johnson, E.L., Padberg, M.W.: Solving large-scale zero-one linear programming problems. Oper. Res. 31, 803–834 (1983)

    Article  MATH  Google Scholar 

  24. Espinoza, D., Fukasawa, R., Goycoolea, M.: Lifting, tilting and fractional programming revisited. Oper. Res. Lett. 559–563 (2010)

  25. Espinoza, D.G.: On Linear Programming, Integer Programming and Cutting Planes, PhD thesis, School of Industrial and Systems Engineering. Georgia Institute of Technology (2006)

  26. Gomory, R.E.: An algorithm for the mixed integer problem. Tech. Rep. RM-2597, RAND Corporation (1960)

  27. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, 2nd edn. Springer, Berlin (1993)

  28. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted cover inequalities for 0–1 integer programs. Math. Program. 85, 437–467 (1999)

    Article  MathSciNet  Google Scholar 

  29. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Sequence independent lifting in mixed integer programming. J. Combinator. Optim. 4, 109–129 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. ILOG: User’s Manual, ILOG CPLEX 10.0, ILOG CPLEX Division, Incline Village, Nevada (2006)

  31. Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R., Danna, E., Gamrath, G., Gleixner, A., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D., Wolter, K.: MIPLIB 2010. Math. Program. Comput. 3(2011), 103–163 (2011). doi:10.1007/s12532-011-0025-9

    Article  MathSciNet  Google Scholar 

  32. Lenstra H.W Jr.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    Google Scholar 

  33. Linderoth, J.T., Savelsbergh, M.W.P.: A computational study of search strategies for mixed integer programming. INFORMS J. Comput. 11, 173–187 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2003)

    Article  MathSciNet  Google Scholar 

  35. Mittelmann, H.: Mixed integer linear programming benchmark (free codes). http://plato.asu.edu/ftp/milpf.html (2008)

  36. Perregaard, M., Balas, E.: Generating cuts from multiple-term disjunctions. In: IntegerProgramming and Combinatorial Optimization, 8th International IPCO Conference, Utrecht, The Netherlands, pp. 348–360. Springer, Berlin (2001)

  37. Zanette, A., Fischetti, M., Balas, E.: Lexicography and degeneracy: can a pure cutting plane algorithm work? Math. Program. 1–24 (2009). doi:10.1007/s10107-009-0335-0

Download references

Acknowledgments

We would like to thank the anonymous referees for their comments and suggestions, which greatly improved the readability and quality of this manuscript. We would also like to give special thanks to Egon Balas and Anureet Saxena for making available their detailed results and actual split cuts for all MIPLIB 3.0 instances, as presented in [12].

Author information

Authors and Affiliations

  1. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

    Vašek Chvátal

  2. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, USA

    William Cook

  3. Department of Industrial Engineering, Universidad de Chile, Santiago, Chile

    Daniel Espinoza

Authors
  1. Vašek Chvátal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William Cook
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Espinoza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Espinoza.

Additional information

W. Cook was supported by NSF Grant CMMI-0726370 and ONR Grant N00014-03-1-0040. D. Espinoza was supported by FONDECYT Grant 1110024 and ICM Grant P10-024-F.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 201 KB)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chvátal, V., Cook, W. & Espinoza, D. Local cuts for mixed-integer programming. Math. Prog. Comp. 5, 171–200 (2013). https://doi.org/10.1007/s12532-013-0052-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-013-0052-9

Mathematics Subject Classification

Search

Navigation