Skip to main content

Advertisement

Log in

Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper, we study the relationship between 2D lattice-free cuts, the family of cuts obtained by taking two-row relaxations of a mixed-integer program (MIP) and applying intersection cuts based on maximal lattice-free sets in \({\mathbb{R}^2}\), and various types of disjunctions. Recently Li and Richard (2008), studied disjunctive cuts obtained from t-branch split disjunctions of mixed-integer sets (these cuts generalize split cuts). Balas (Presentation at the Spring Meeting of the American Mathematical Society (Western Section), San Francisco, 2009) initiated the study of cuts for the two-row continuous group relaxation obtained from 2-branch split disjunctions. We study these cuts (and call them cross cuts) for the two-row continuous group relaxation, and for general MIPs. We also consider cuts obtained from asymmetric 2-branch disjunctions which we call crooked cross cuts. For the two-row continuous group relaxation, we show that unimodular cross cuts (the coefficients of the two split inequalities form a unimodular matrix) are equivalent to the cuts obtained from maximal lattice-free sets other than type 3 triangles. We also prove that all 2D lattice-free cuts and their S-free extensions are crooked cross cuts. For general mixed integer sets, we show that crooked cross cuts can be generated from a structured three-row relaxation. Finally, we show that for the corner relaxation of an MIP, every crooked cross cut is a 2D lattice-free cut.

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.

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: Fischetti, M., Williamson, D.P. (eds.) Proceedings 12th Conference on Integer Programming and Combinatorial Optimization (LNCS 4513), pp. 1–15. Springer-Verlag (2007)

  2. Andersen K., Louveaux Q., Weismantel R.: Mixed-integer sets from two rows of two adjacent simplex bases. Math. Program. 124, 455–480 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Balas E.: Disjunctive programming. Ann. Discrete Math. 5, 3–51 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balas, E.: Intersection cuts from maximal lattice-free convex sets and lift-and-project cuts from multiple-term disjunctions. In: Presentation at the Spring Meeting of the American Mathematical Society (Western Section). San Francisco (2009)

  6. Balas E., Jeroslow R.: Strenghtening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: On the relative strength of split, triangle, and quadrilateral cuts. Math. Program. (2010, to appear)

  8. Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: Experiments with two row cuts from degenerate tableaux. INFORMS J. Comput. (2010, to appear)

  9. Basu, A., Campelo, M., Conforti, M., Cornuéjols, G., Zambelli, G.: On lifting integer variables in minimal inequalities. In: Eisenbrand, F., Shepherd, B. (eds.) Proceedings 14th Conference on Integer Programming and Combinatorial Optimization (LNCS 6080) pp. 85–95. Springer-Verlag (2010)

  10. Basu A., Conforti M., Cornuéjols G., Zambelli G.: Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discrete Math. 24, 158–168 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Basu A., Conforti M., Cornuéjols G., Zambelli G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Borozan V., Cornuéjols G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Conforti, M., Cornuéjols, G., Zambelli, G.: A geometric perspective on lifting. Oper. Res. (2009, to appear)

  14. Conforti M., Cornuéjols G., Zambelli G.: Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38, 153–155 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Cornuéjols G., Margot F.: On the facets of mixed integer programs with two integer variables and two constraints. Math. Program. 120, 429–456 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Dey S.S., Tramontani A.: Recent developments in multi-row cuts. Optima 80, 28 (2009)

    Google Scholar 

  19. Dey, S.S., Wolsey, L.A.: Two row mixed integer cuts via lifting, Technical Report CORE DP 30, Université catholique de Louvain, Louvain-la-Neuve, Belgium (2008)

  20. Dey S.S., Wolsey L.A.: Two row mixed integer cuts via lifting. Math. Program. 124, 143–174 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dey S.S., Wolsey L.A.: Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20, 2890–2912 (2010a)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dey S.S., Wolsey L.A.: Composite lifting of group inequalities and an application to two-row mixing inequalities. Discrete Optim. 7, 256–268 (2010b)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dey, S.S., Lodi, A., Wolsey, L.A., Tramontani, A.: Experiments with two row tableau cuts. In: Eisenbrand, F., Shepherd, B. (eds.) Proceedings 14th Conference on Integer Programming and Combinatorial Optimization (LNCS 6080), pp. 424–437. Springer-Verlag (2010c)

  24. Espinoza, D.: Computing with multiple-row Gomory cuts. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) Proceedings 13th Conference on Integer Programming and Combinatorial Optimization (LNCS 5035), pp. 214–224. Springer-Verlag (2009)

  25. Fukasawa, R., Günlük, O.: Strengthening lattice-free cuts with nonnegativity. http://www.optimization-online.org/DB_HTML/2009/05/2296.html

  26. Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part II. Math. Program. 3, 359–389 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gomory R.E., Johnson E.L., Evans L.: polyhedra and their connection with cutting planes. Math. Program. 96, 321–339 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Johnson L.E.: Characterization of facets for multiple right-hand side choice linear programs. Math. Program. Study 14, 112–142 (1981)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Li Y., Richard J.-P.P.: Cook, Kannan and Schrijver’s example revisited. Discrete Optim. 5, 724–734 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lovász L.: Geometry of numbers and integer programming. In: Iri, M., Tanabe, K. (eds) Mathematical Programming: Recent Developments and Applications, pp. 177–201. Kluwer, Dordrecht (1989)

    Google Scholar 

  32. Zambelli G.: On degenerate multi-row Gomory cuts. Oper. Res. Lett. 37, 21–22 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sanjeeb Dash.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dash, S., Dey, S.S. & Günlük, O. Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra. Math. Program. 135, 221–254 (2012). https://doi.org/10.1007/s10107-011-0455-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-011-0455-1

Mathematics Subject Classification (2000)

Navigation