Abstract
Following the flurry of recent theoretical work on cutting planes from two row mixed integer group relaxations of an LP tableau, we report on some computational tests to evaluate the effectiveness of two row cuts based on lattice-free (type 2) triangles having more than one integer point on one side. A heuristic procedure to generate such triangles is presented, and then the coefficients of the integer variables are tightened by lifting. As a first step in testing the effectiveness of the triangle cuts, we make comparisons between the gap closed using Gomory mixed integer cuts for one round and the gap closed in one round using all the triangles generated by our heuristic. Our tests are carried out on different classes of randomly generated instances designed to represent different models in the literature by varying the number of integer non-basic variables, bounds and non-negativity constraints.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Operations Research Letters 34, 361–372 (2006), http://miplib.zib.de
Andersen, K., Cornuéjols, G., Li, Y.: Reduce-and-split cuts: Improving the performance of mixed integer Gomory cuts. Management Science 51, 1720–1732 (2005)
Andersen, K., Louveaux, Q., Weismantel, R.: An analysis of mixed integer linear sets based on lattice point free convex sets. Mathematics of Operations Research 35, 233–256 (2010)
Andersen, K., Louveaux, Q., Weismantel, R.: Mixed-integer sets from two rows of two adjacent simplex bases (2009), http://hdl.handle.net/2268/35089
Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.A.: Cutting planes from two rows of a simplex tableau. In: Fischetti, M., Williamson, D.P. (eds.) Proceedings 12th Conference on Integer and Combinatorial Optimization, pp. 1–15. Springer, Heidelberg (2007)
Atamturk, A.: http://ieor.berkeley.edu/~atamturk/data/
Balas, E.: Intersection cuts - a new type of cutting planes for integer programming. Operations Research 19, 19–39 (1971)
Balas, E., Jeroslow, R.: Strenghtening cuts for mixed integer programs. European Journal of Operations Research 4, 224–234 (1980)
Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces (2009), http://www.math.unipd.it/~giacomo/
Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Minimal inequalities for an infinite relaxation of integer programs (2009), http://www.math.unipd.it/~giacomo/
Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Mathematics of Operations Research 34, 538–546 (2009)
Conforti, M., Cornuéjols, G., Zambelli, G.: A geometric perspective on lifting (2009), http://www.math.unipd.it/~giacomo/
Cornuéjols, G., Margot, F.: On the facets of mixed integer programs with two integer variables and two constraints. Mathematical Programming 120, 429–456 (2009)
Dey, S.S., Tramontani, A.: Recent developments in multi-row cuts. Optima 80, 2–8 (2009)
Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) Proceedings 13th Conference on Integer and Combinatorial Optimization, pp. 463–475. Springer, Heidelberg (2008)
Dey, S.S., Wolsey, L.A.: Constrained infinite group relaxations of MIPs, Tech. Report CORE DP 33, Université catholique de Louvain, Louvain-la-Neuve, Belgium (2009)
Espinoza, D.: Computing with multiple-row Gomory cuts. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 214–224. Springer, Heidelberg (2008)
Fischetti, M., Saturni, C.: Mixed integer cuts from cyclic groups. Mathematical Programming 109, 27–53 (2007)
Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–308. Mcgraw-Hill Book Company Inc., New York (1963)
Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, part I. Mathematical Programming 3, 23–85 (1972)
Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, part II. Mathematical Programming 3, 359–389 (1972)
Gomory, R.E., Johnson, E.L.: T-space and cutting planes. Mathematical Programming 96, 341–375 (2003)
Johnson, E.L.: On the group problem for mixed integer programming. Mathematical Programming Study 2, 137–179 (1974)
Lovász, L.: Geometry of numbers and integer programming. Mathematical Programming: Recent Developments and Applications, 177–210 (1989)
Nemhauser, G.L., Wolsey, L.A.: A recursive procedure to generate all cuts for 0-1 mixed integer programs. Mathematical Programming 46, 379–390 (1990)
Zambelli, G.: On degenerate multi-row Gomory cuts. Operations Research Letters 37, 21–22 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dey, S.S., Lodi, A., Tramontani, A., Wolsey, L.A. (2010). Experiments with Two Row Tableau Cuts. In: Eisenbrand, F., Shepherd, F.B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2010. Lecture Notes in Computer Science, vol 6080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13036-6_32
Download citation
DOI: https://doi.org/10.1007/978-3-642-13036-6_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13035-9
Online ISBN: 978-3-642-13036-6
eBook Packages: Computer ScienceComputer Science (R0)