Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

On the facets of mixed integer programs with two integer variables and two constraints

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables, nonnegative continuous variables and two equality constraints, and we give a complete characterization of its facets. We also derive an analogous characterization of the facets of the underlying finite integer program.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.: Cutting planes from two rows of a simplex tableau. In: Proceedings of IPCO XII. Ithaca, New York, pp 1–15 (2007)

  2. Balas E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bell D.E.: A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)

    Google Scholar 

  4. Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints, technical report (2007)

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Dey, S.S., Richard, J.-P.P., Miller, L.A., Li, Y.: Extreme inequalities for infinite group problems, technical report (2006)

  8. Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. IPCO XIII, Bertinoro, Italy (2008)

  9. Gomory R.E.: Some polyhedra related to combinatorial problems. J. Lin. Algebra Appl. 2, 451–558 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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 

  11. Gomory, R.E.: Thoughts about integer programming. In: 50th anniversary symposium of OR, University of Montreal, January 2007, and Corner polyhedra and two-equation cutting planes. In: George Nemhauser symposium, Atlanta, (2007)

  12. Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part I. Math Program 3, 23–85 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Johnson, E.L.: On the group problem for mixed integer programming. In: Mathematical programming study, pp 137–179 (1974)

  14. Lovász L.: Geometry of numbers and integer programming. In: Iri, M., Tanabe, K.(eds) Mathematical programming: recent developments and applications., pp. 177–210. Kluwer, Dordrecht (1989)

    Google Scholar 

  15. Meyer, C.D.: Matrix analysis and applied linear algebra. SIAM (2000)

  16. Meyer R.R.: On the existence of optimal solutions to integer and mixed integer programming problems. Math. Program. 7, 223–235 (1974)

    Article  MATH  Google Scholar 

  17. Nemhauser G.L., Wolsey L.A.: Integer and combinatorial optimization. Wiley, New York (1988)

    MATH  Google Scholar 

  18. Scarf H.E.: An observation on the structure of production sets with indivisibilities. Proc. Nat. Acad. Sci. USA 74, 3637–3641 (1977)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols & François Margot

  2. LIF, Faculté des Sciences de Luminy, Université de Marseille, Marseille, France

    Gérard Cornuéjols

Authors
  1. Gérard Cornuéjols
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. François Margot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard Cornuéjols.

Additional information

Gérard Cornuéjols was supported by NSF grant CMMI0653419, ONR grant N00014-97-1-0196 and ANR grant BLAN06-1-138894.

François Margot was supported by ONR grant N00014-97-1-0196.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cornuéjols, G., Margot, F. On the facets of mixed integer programs with two integer variables and two constraints. Math. Program. 120, 429–456 (2009). https://doi.org/10.1007/s10107-008-0221-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-008-0221-1

Keywords

Search

Navigation