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Mathematical Programming Computation

, Volume 11, Issue 2, pp 211–235 | Cite as

The (not so) trivial lifting in two dimensions

  • Ricardo Fukasawa
  • Laurent Poirrier
  • Álinson S. XavierEmail author
Full Length Paper
  • 89 Downloads

Abstract

When generating cutting-planes for mixed-integer programs from multiple rows of the simplex tableau, the usual approach has been to relax the integrality of the non-basic variables, compute an intersection cut, then strengthen the cut coefficients corresponding to integral non-basic variables using the so-called trivial lifting procedure. Although of polynomial-time complexity in theory, this lifting procedure can be computationally costly in practice. For the case of two-row relaxations, we present a practical algorithm that computes trivial lifting coefficients in constant time, for arbitrary maximal lattice-free sets. Computational experiments confirm that the algorithm works well in practice.

Keywords

Integer programming Lifting Cutting planes 

Mathematics Subject Classification

90C11 90C57 

Supplementary material

References

  1. 1.
    Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.: Inequalities from two rows of a simplex tableau. In: Fischetti, M., Williamson, D. (eds.) Integer Programming and Combinatorial Optimization, volume 4513 of Lecture Notes in Computer Science, pp. 1–15. Springer, Berlin (2007)Google Scholar
  2. 2.
    Averkov, G., Basu, A.: Lifting properties of maximal lattice-free polyhedra. Math. Program. 154(1–2), 81–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balas, E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 1(19), 19–39 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balas, E., Jeroslow, R.G.: Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4(4), 224–234 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: Experiments with two-row cuts from degenerate tableaux. INFORMS J. Comput. 23, 578–590 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Basu, A., Campêlo, M., Conforti, M., Cornuéjols, G., Zambelli, G.: Unique lifting of integer variables in minimal inequalities. Math. Program. 141(1–2), 561–576 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming, vol. 271. Springer, Berlin (2014)zbMATHGoogle Scholar
  9. 9.
    Dey, S.S., Lodi, A., Tramontani, A., Wolsey, L.A.: On the practical strength of two-row tableau cuts. INFORMS J. Comput. 26(2), 222–237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dey, S.S., Louveaux, Q.: Split rank of triangle and quadrilateral inequalities. Math. Oper. Res. 36(3), 432–461 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dey, S.S., Wolsey, L.A.: Two row mixed-integer cuts via lifting. Math. Program. 124, 143–174 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Espinoza, D.G.: Computing with multi-row Gomory cuts. Oper. Res. Lett. 38(2), 115–120 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fukasawa, R., Poirrier, L., Xavier, Á.S.: The (not so) trivial lifting in two dimensions: source code, Aug 2018.  https://doi.org/10.5281/zenodo.1342770
  14. 14.
    Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2(4), 451–558 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, part I. Math. Program. 3, 23–85 (1972)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Sequence independent lifting in mixed integer programming. J. Comb. Optim. 4, 109–129 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hurkens, C.A.J.: Blowing up convex sets in the plane. Linear Algebra Appl. 134, 121–128 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kannan, R., Lovász, L.: Covering Minima and Lattice Point Free Convex Bodies, pp. 193–213. Springer, Berlin (1986)zbMATHGoogle Scholar
  19. 19.
    Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math. Program. Comput. 3(2), 103–163 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Louveaux, Q., Poirrier, L.: An algorithm for the separation of two-row cuts. Math. Program. 143(1–2), 111–146 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Louveaux, Q., Poirrier, L., Salvagnin, D.: The strength of multi-row models. Math. Program. Comput. 7(2), 113–148 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Oertel, T., Wagner, C., Weismantel, R.: Convex integer minimization in fixed dimension. http://arxiv.org/pdf/1203.4175v1.pdf (2012). Accessed 9 Aug 2018
  24. 24.
    Padberg, M.W.: On the facial structure of set packing polyhedra. Math. Program. 5(1), 199–215 (1973)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  • Ricardo Fukasawa
    • 1
  • Laurent Poirrier
    • 1
  • Álinson S. Xavier
    • 2
    Email author
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Energy Systems DivisionArgonne National LaboratoryLemontUSA

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