Mathematical Programming Computation

, Volume 2, Issue 3–4, pp 231–257 | Cite as

A heuristic to generate rank-1 GMI cuts

  • Sanjeeb Dash
  • Marcos Goycoolea
Full Length Paper


Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts.

Mathematics Subject Classification (2000)

90C11 90C57 


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  1. 1.
    Achterberg T., Kock T., Martin A.: MIPLIB 2003. Oper. Res. Lett. 34(4), 361–372 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Andersen K., Cornuéjols G., Li Y.: Reduce-and-split cuts: improving the performance of mixed integer gomory cuts. Manage. Sci. 51, 1720–1732 (2005)CrossRefGoogle Scholar
  3. 3.
    Applegate, D., Cook, W., Dash, S., Mevenkamp, M.: QSopt linear programming solver. (2004)
  4. 4.
    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, 165–200 (2010)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Balas E., Ceria S., Cornuejols G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Balas E., Ceria S., Cornuejols G., Natraj N.: Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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. Program. 94, 221–245 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Balas E., Saxena A.: Optimizing over the split closure. Math. Program. 113, 219–240 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
  10. 10.
    Bonami P., Minoux M.: Using rank-1 lift-and-project closures to generate cuts for 0–1 MIPs, a computational investigation. Discret. Optim. 2, 288–307 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bixby, R., Gu, Z., Rothberg, E., Wunderling, R.: The sharpest cut: the impact of Manfred Padberg and his work, chap. 18 (mixed-integer programming: a progress report), pp. 309–323. MPS-SIAM Series on Optimization (2004)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Bixby, R.E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: Mip: theory and practice—closing the gap. In: System modelling and optimization, pp. 19–50 (1999)Google Scholar
  14. 14.
    Bussieck M.R., Ferris M.C., Meeraus A.: Grid-enabled optimization with GAMS. INFORMS J. Comput. 21(3), 349–362 (2009)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Caprara A., Letchford A.: On the separation of split cuts and related inequalities. Math. Program. 94, 279–294 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ceria, S., Cornuéjols, G., Dawande, M.: Combining and strengthening Gomory cuts. In: Balas, E., Clausen, J. (eds) Proceedings of IPCO 1995. Lect. Notes Comput. Sci. 920, 438–451 (1995)Google Scholar
  17. 17.
    COIN-OR, The computational infrastructure for operations research project.
  18. 18.
    Cook W., Dash S., Goycoolea M., Fukasawa R.: Numerically safe Gomory mixed-integer cuts. INFORMS J. Comput. 21, 641–649 (2009)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Cornuéjols G., Li Y.: Elementary closures for integer programs. Oper. Res. Lett. 28, 1–8 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Dash S., Günlük O.: On the strength of Gomory mixed-integer cuts as group cuts. Math. Program. 115, 387–407 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dash S., Goycoolea M., Günlük O.: Two-step mir inequalities for mixed-integer programs. INFORMS J. Comput. 22, 236–249 (2010)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Dash S., Günlük O., Lodi A.: MIR closures of polyhedral sets. Math. Program. 121(1), 33–60 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Edmonds J.: Matroids and the greedy algorithm. Math. Program. 1(1), 127–136 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Fischetti M., Lodi A.: Optimizing over the first Chvátal closure. Math. Program. B 110(1), 3–20 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Fischetti M., Saturni C.: Mixed integer cuts from cyclic groups. Math. Program. 109, 27–53 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Fukasawa, R., Goycoolea, M.: On the exact separation of mixed-integer knapsack cuts. In: Fischetti, M., Williamson, D.P. (eds) Proceedings of IPCO 2007. Lect. Notes Comput. Sci. 4513, 225–239 (2007)Google Scholar
  27. 27.
    Geoffrion A.M., Graves G.W.: Multicommodity distribution system design by Benders decomposition. Manage. Sci. 20, 822–844 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Gomory, R.E.: An algorithm for the mixed integer problem. RM-2597, The Rand Corporation (1960)Google Scholar
  29. 29.
    Liebchen, C., Moehring, R.H.: Information on the MIPLIB’s timetab-instances. Technical Report 2003/49, Department of Mathematics, Technical University Berlin (2003)Google Scholar
  30. 30.
    Marchand H., Wolsey L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Nemhauser G.L., Wolsey L.A.: Integer and combinatorial optimization. Wiley, New York (1988)zbMATHGoogle Scholar
  32. 32.
    Margot F.: Testing cut generators for mixed-integer linear programming. Math. Program. Comput. 1, 69–95 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Markowitz H.M.: The elimination from of inverse and its applications to linear programming. Manage. Sci. 3, 255–269 (1957)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Mittelmann, H.: MIQPlib.
  35. 35.
    Pinar A., Chow E., Pothen A.: Combinatorial algorithms for computing column space bases that have sparse inverses. Electron. Trans. Numer. Anal. 22, 122–145 (2006)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Suhl U.H., Suhl L.M.: Computing sparse LU factorizations for large-scale linear programming bases. ORSA J. Comput. 2(4), 325–335 (1990)zbMATHGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2010

Authors and Affiliations

  1. 1.Mathematical Programming GroupIBM T. J. Watson Research CenterYorktown HeightsUSA
  2. 2.School of BusinessUniversidad Adolfo IbáñezSantiagoChile

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