Abstract
In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of these variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. An extensive computational evaluation of these cuts points to the following two conclusions. The first is that our rank-1 cuts improve significantly on existing split cut generators (Gomory cuts from single tableau rows, MIR, Reduce-and-Split, Lift-and-Project, Flow and Knapsack cover): on MIPLIB instances, these generators close 24% of the integrality gap on average; adding our cuts yields an additional 5%. The second conclusion is that, when incorporated in a Branch-and-Cut framework, these new cuts can improve computing time on difficult instances.
Similar content being viewed by others
References
Achterberg T., Koch T., Martin A.: MIPLIB 2003. Oper. Res. Lett. 34(4), 361–372 (2006)
Ajtai, M.: The shortest vector problem in l 2 is NP-hard for randomized reductions. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing. Dallas, TX (1998)
Andersen K., Cornuéjols G., Li Y.: Reduce-and-split cuts: improving the performance of mixed integer gomory cuts. Manag. Sci. 51(11), 1720–1732 (2005)
Balas E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19(1), 19–39 (1971)
Balas E.: Disjunctive programming. Annal. Discret. Math. 5, 3–51 (1979)
Balas E., Bonami P.: Generating Lift-and-Project cuts from the LP simplex tableau: open source implementation and testing of new variants. Math. Program. Comput. 1, 165–199 (2009)
Balas E., Jeroslow R.G.: Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)
Balas E., Saxena A.: Optimizing over the split closure. Math. Program. 113(2), 219–240 (2008)
Bixby R., Rothberg E.: Progress in computational mixed integer programming—a look back from the other side of the tipping point. Annal. Oper. Res. 149(1), 37–41 (2007)
Bixby R.E., Ceria S., McZeal C.M., Savelsbergh M.W.P.: An updated mixed integer programming library: MIPLIB 3.0. Optima 58, 12–15 (1998)
Bonami P., Cornuéjols G., Dash S., Fischetti M., Lodi A.: Projected Chvátal-Gomory cuts for mixed integer linear programs. Math. Program. 113, 241–257 (2008)
COIN-OR Branch-and-Cut. https://projects.coin-or.org/Cbc. Accessed Oct 2010
COIN-OR Cut Generation Library. https://projects.coin-or.org/Cgl. Accessed Oct 2010
COIN-OR Linear Programming. https://projects.coin-or.org/Clp. Accessed Oct 2010
Cook W., Kannan R., Schrijver A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)
Cornuéjols, G., Liberti, L., Nannicini, G.: Improved strategies for branching on general disjunctions. Math. Program. A (2009). Published online
Dash S., Günlük O., Lodi A.: MIR closures of polyhedral sets. Math. Program. 121(1), 33–60 (2010)
Gomory R.E.: An algorithm for the mixed-integer problem. Tech. Rep. RM-2597, RAND Corporation, (1960)
Lenstra A.K., Lenstra H.W. Jr, Lovász L.: Factoring polynomials with rational coefficients. Mathematische Annalen 4(261), 515–534 (1982)
Margot F.: Testing cut generators for mixed-integer linear programming. Math. Program. Comput. 1(1), 69–95 (2009)
Nemhauser G.L., Wolsey L.: A recursive procedure for generating all cuts for 0-1 mixed integer programs. Math. Program. 46, 379–390 (1990)
Papadimitriou C., Steiglitz K.: Combinatorial Optimization: Algorithms and Complexity. Dover, New York (1990)
Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes in C, Second Edition. Cambridge University Press, Cambridge (1992, reprinted 1997)
Wolsey L.: Integer Programming. Wiley, New York (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cornuéjols, G., Nannicini, G. Practical strategies for generating rank-1 split cuts in mixed-integer linear programming. Math. Prog. Comp. 3, 281–318 (2011). https://doi.org/10.1007/s12532-011-0028-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12532-011-0028-6