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.
Similar content being viewed by others
References
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)
Andersen K., Louveaux Q., Weismantel R.: Mixed-integer sets from two rows of two adjacent simplex bases. Math. Program. 124, 455–480 (2010)
Balas E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)
Balas E.: Disjunctive programming. Ann. Discrete Math. 5, 3–51 (1979)
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)
Balas E., Jeroslow R.: Strenghtening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)
Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: On the relative strength of split, triangle, and quadrilateral cuts. Math. Program. (2010, to appear)
Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: Experiments with two row cuts from degenerate tableaux. INFORMS J. Comput. (2010, to appear)
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)
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)
Basu A., Conforti M., Cornuéjols G., Zambelli G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)
Borozan V., Cornuéjols G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)
Conforti, M., Cornuéjols, G., Zambelli, G.: A geometric perspective on lifting. Oper. Res. (2009, to appear)
Conforti M., Cornuéjols G., Zambelli G.: Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38, 153–155 (2010)
Cook W.J., Kannan R., Schrijver A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)
Cornuéjols G., Margot F.: On the facets of mixed integer programs with two integer variables and two constraints. Math. Program. 120, 429–456 (2009)
Cornuéjols G., Li Y.: On the rank of mixed 0,1 polyhedra. Math. Program. 91, 391–397 (2002)
Dey S.S., Tramontani A.: Recent developments in multi-row cuts. Optima 80, 28 (2009)
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)
Dey S.S., Wolsey L.A.: Two row mixed integer cuts via lifting. Math. Program. 124, 143–174 (2010)
Dey S.S., Wolsey L.A.: Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20, 2890–2912 (2010a)
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)
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)
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)
Fukasawa, R., Günlük, O.: Strengthening lattice-free cuts with nonnegativity. http://www.optimization-online.org/DB_HTML/2009/05/2296.html
Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part II. Math. Program. 3, 359–389 (1972)
Gomory R.E., Johnson E.L., Evans L.: polyhedra and their connection with cutting planes. Math. Program. 96, 321–339 (2003)
Johnson L.E.: Characterization of facets for multiple right-hand side choice linear programs. Math. Program. Study 14, 112–142 (1981)
Nemhauser G., Wolsey L.A.: A recursive procedure to generate all cuts for 0–1 mixed integer programs. Math. Program. 46, 379–390 (1990)
Li Y., Richard J.-P.P.: Cook, Kannan and Schrijver’s example revisited. Discrete Optim. 5, 724–734 (2008)
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)
Zambelli G.: On degenerate multi-row Gomory cuts. Oper. Res. Lett. 37, 21–22 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-011-0455-1