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On optimizing over lift-and-project closures

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Abstract

The strengthened lift-and-project closure of a mixed integer linear program is the polyhedron obtained by intersecting all strengthened lift-and-project cuts obtained from its initial formulation, or equivalently all mixed integer Gomory cuts read from all tableaux corresponding to feasible and infeasible bases of the LP relaxation. In this paper, we present an algorithm for approximately optimizing over the strengthened lift-and-project closure. The originality of our method is that it relies on a cut generation linear programming problem which is obtained from the original LP relaxation by only modifying the bounds on the variables and constraints. This separation LP can also be seen as dual to the cut generation LP used in disjunctive programming procedures with a particular normalization. We study properties of this separation LP, and discuss how to use it to approximately optimize over the strengthened lift-and-project closure. Finally, we present computational experiments and comparisons with recent related works.

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References

  1. Andersen K., Cornuéjols G., Li Y.: Reduce-and-split cuts: improving the performance of mixed integer Gomory cuts. Manag. Sci. 50(11), 1720–1732 (2005)

    Article  Google Scholar 

  2. Andersen K., Cornuéjols G., Li Y.: Split closure and intersection cuts. Math. Progr. 102, 457–493 (2005)

    Article  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 and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebraic Discret. Methods 6(3), 466–486 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balas, E.: Disjunctive programming: Properties of the convex hull of feasible points. Discret. Appl. Math. 89, 3–44 (1998) (originally MSRR # 348, Carnegie Mellon University, July 1974)

    Google Scholar 

  6. Balas E., Bonami P.: Generating lift-and-project cuts from the LP simplex tableau: open source implementation and testing of new variants. Math. Progr. Comput. 1, 165–199 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balas E., Ceria S., Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Progr. 58, 295–324 (1993)

    Article  MATH  Google Scholar 

  8. Balas E., Ceria S., Cornuéjols G.: Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Manag. Sci. 42, 1229–1246 (1996)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Balas E., Perregaard M.: Lift and project for mixed 0-1 programming: recent progress. Discret. Appl. Math. 123, 129–154 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Balas E., Perregaard M.: A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0-1 programming. Math. Progr. 94, 221–245 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Balas E., Saxena A.: Optimizing over the split closure. Math. Progr. 113, 219–240 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Balas E., Zemel E.: Facets of the knapsack polytope from minimal covers. SIAM J. Appl. Math. 34, 119–148 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bixby, R., Ceria, S., McZeal, C., Savelsbergh, M.: Miplib 3.0. http://www.caam.rice.edu/~bixby/miplib/miplib.html (1998)

  15. Bixby, R., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: The Sharpest cut, chapter mixed-integer programming: a progress report, pp. 309–326. MPS-SIAM Series on Optimization. SIAM (2004)

  16. Bonami, P., Balas, E.: Cgllandp. https://projects.coin-or.org/cgl/wiki/cgllandp. July (2006)

  17. Bonami P., Cornuéjols G., Dash S., Fischetti M., Lodi A.: Projected Chvátal–Gomory cuts for mixed integer linear programs. Math. Progr. Series A 113(2), 241–257 (2008)

    Article  MATH  Google Scholar 

  18. Bonami P., Minoux M.: Using rank-1 lift-and-project closures to generate cuts for 0–1 MIPs, a computational investigation. Discret. Optim. 2(4), 288–307 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Caprara A., Letchford A.N.: On the separation of split cuts and related inequalities. Math. Progr. 94(2–3), 279–294 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chvátal V.: Edmonds polytopes and a hierarchy of combinatorial optimization. Discret. Math. 4, 305–337 (1973)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Cornuéjols G., Li Y.: Elementary closures for integer programs. Oper. Res. Lett. 28, 1–8 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cornuéjols, G., Nannicini, G. Practical strategies for generating rank-1 split cuts in mixed-integer linear programming. Math. Progr. Comput. pp. 1–38. doi:10.1007/s12532-011-0028-6

  24. Dash S., Goycoolea M.: A heuristic to generate rank-1 gmi cuts. Math. Progr. Comput. 2, 231–257 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dash S., Günlük O., Lodi A.: MIR closures of polyhedral sets. Math. Progr. 121(1), 33–60 (2010)

    Article  MATH  Google Scholar 

  26. Eisenbrand, F.: On the membership problem for the elementary closure of a polyhedron. Combinatorica 19(2), (1999)

  27. Fischetti M., Lodi A.: Optimizing over the first Chvátal closure. Math. Progr. 110, 3–20 (2007). doi:10.1007/s10107-006-0054-8

    Article  MathSciNet  MATH  Google Scholar 

  28. Fischetti M., Lodi A., Tramontani A.: On the separation of disjunctive cuts. Math. Progr. 128, 205–230 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fischetti, M., Salvagnin, D.: An in-out approach to disjunctive optimization. In: Lodi, A., Milano, M., Toth, P., (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. Lecture Notes in Computer Science, vol. 6140, pp. 136–140. Springer, Berlin (2010)

  30. Fischetti M., Salvagnin D.: A relax-and-cut framework for gomory’s mixed-integer cuts. Math. Progr. Comput. 3, 79–102 (2011)

    Article  MathSciNet  Google Scholar 

  31. Forrest, J.: CLP. http://www.coin-or.org/ (2004)

  32. Gomory R.: An algorithm for integer solution solutions to linear programming. In: Graves, R.L., Wolfe, P. (eds) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)

    Google Scholar 

  33. Gomory, R.E.: Solving linear programming problems in integers. In: Bellman R., Hall, M. (eds.) Combinatorial Analysis, Proceedings of Symposia in Applied Mathematics, vol. 10, pp. 211–216. Providence (1960)

  34. Kelley J.E.: The cutting plane method for solving convex programs. J. SIAM 8(4), 703–712 (1960)

    MathSciNet  Google Scholar 

  35. Lougee-Heimer, R.: The common optimization interface for operations research. IBM J. Res. Dev. 47, 57–66 (2003). http://www.coin-or.org

  36. Martin, A., Achterberg, T., Koch, T.: MIPLIB 2003. http://miplib.zib.de (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Padberg M., Roy T., Wolsey L.: Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  39. Roy T.V., Wolsey L.: Solving mixed integer programming problems using automatic reformulation. Oper. Res. 35, 45–57 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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  1. LIF, CNRS, Aix Marseille University, 163 Avenue de Luminy - Case 901, 13008, Marseille, France

    Pierre Bonami

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Correspondence to Pierre Bonami.

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Bonami, P. On optimizing over lift-and-project closures. Math. Prog. Comp. 4, 151–179 (2012). https://doi.org/10.1007/s12532-012-0037-0

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