Skip to main content
Log in

MIPping Closures: An Instant Survey

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

A relevant amount of work has been devoted very recently to modeling and (heuristically) solving the NP-hard separation problem of famous classes of valid inequalities for mixed integer linear programs (MIPs). This task has been accomplished by using, in turn, mixed-integer linear models for the separation problem, and a general-purpose solver to actually find violated cuts—the so-called MIPping approach. We instantly survey these attempts by discussing their computational outcome and pointing out their practical interest for future integration in the MIP solvers.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Achterberg, T., Koch, T., Martin, A.: The mixed integer programming library: MIPLIB 2003. http://miplib.zib.de

  • Balas, E., Perregaard, M.: Lift and project for mixed 0–1 programming: recent progress. Discrete Appl. Math. 123, 129–154 (2002)

    Google Scholar 

  • Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. (2006). DOI 10.1007/s10107-006-0049-5

  • Bixby, R. E., Ceria, S., McZeal, C. M., Savelsbergh, M. W. P.: MIPLIB 3.0. http:// www.caam.rice.edu/~bixby/miplib/miplib.html

  • Bonami, P., Cornuéjols, G., Dash, S., Fischetti, M., Lodi, A.: Projected Chvátal-Gomory cuts for mixed integer linear programs. Math. Program. (2006, in press). DOI: 10.1007/s10107-006-0051-y

  • Caprara, A., Letchford, A. N.: On the separation of split cuts and related inequalities. Math. Program. 94, 279–294 (2003)

    Google Scholar 

  • Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4, 305–337 (1973)

    Google Scholar 

  • Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)

    Google Scholar 

  • Cornuéjols, G.: Valid inequalities for mixed integer linear programs. Math. Program. (2006, in press). DOI: 10.1007/s10107-006-0086-0

  • Cornuéjols, G., Li, Y.: On the rank of mixed 0,1 polyhedra. Math. Program. 91, 391–397 (2002)

    Google Scholar 

  • Dash, S., Günlük, O., Lodi, A.: On the MIR closure of polyhedra. In: Fischetti M., Williamson, D.P. (eds.) Integer Programming and Combinatorial Optimization-IPCO 2007, Lecture Notes in Computer Science, Springer (2007, in press)

  • De Franceschi, R., Fischetti, M., Toth, P.: A new ILP-based refinement heuristic for vehicle routing problems. Math. Program. 105, 471–499 (2006)

    Google Scholar 

  • Eisenbrand, F.: On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)

    Google Scholar 

  • Fischetti, M., Lodi, A.: Local branching. Math. Program. 98, 23–47 (2003)

  • Fischetti, M., Lodi, A.: Optimizing over the first Chvátal closure. In: Jünger, M., Kaibel, V. (eds.) Integer Programming and Combinatorial Optimization. Vol. 3509. pp. 12–22. IPCO 2005, LNCS, Springer, Berlin Heidelberg (2005)

  • Gomory, R. E.: Outline of an algorithm for integer solutions to linear programs. Bull. AMS 64, 275–278 (1958)

    Google Scholar 

  • 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)

  • Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)

  • Nazareth, J. L.: The homotopy principle and algorithms for linear programming. SIAM J. Optim. 1, 316–332 (1991)

    Google Scholar 

  • Nemhauser, G. L., Wolsey, L. A.: A recursive procedure to generate all cuts for 0-1 mixed integer programs. Math. Program. 46, 379–390 (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrea Lodi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fischetti, M., Lodi, A. MIPping Closures: An Instant Survey. Graphs and Combinatorics 23 (Suppl 1), 233–243 (2007). https://doi.org/10.1007/s00373-007-0711-6

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-007-0711-6

Keywords

Navigation