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Two row mixed-integer cuts via lifting

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Abstract

Recently Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.

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References

  1. 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 Programming and Combinatorial Optimization, pp. 1–15. Springer (2007)

  2. Aráoz J., Evans L., Gomory R.E., Johnson E.L.: Cyclic group and knapsack facets. Math. Prog. 96, 377–408 (2003)

    Article  MATH  Google Scholar 

  3. Balas E.: Disjunctive programming. Ann. Discrete Math. 5, 3–51 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balas E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. http://integer.tepper.cmu.edu (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Cornuéjols, G., Margot, F.: On the facets of mixed integer programs with two integer variables and two constraints. To appear in Math. Prog. (2008)

  10. Cornuéjols G., Li Y., Vandenbussche D.: K-cuts: a variation of Gomory mixed integer cuts from the LP tableau. INFORMS J. Comput. 15, 385–396 (2003)

    Article  MathSciNet  Google Scholar 

  11. Dash S., Günlük O.: Valid inequalities based on simple mixed-integer sets. Math. Prog. 105, 29–53 (2006)

    Article  MATH  Google Scholar 

  12. Dash, S., Günlük, O.: On the strength of Gomory mixed-integer cuts as group cuts. Technical Report RC23967, IBM Research report (2006b)

  13. Dey, S.S., Richard, J.-P.P.: Sequential-merge facets for two-dimensional group problems. In: Fischetti, M., Williamson, D.P. (eds.) Proceedings 12th Conference on Integer Programming and Combinatorial Optimization, pp. 30–42. Springer (2007)

  14. Dey S.S., Richard J.-P.P.: Facets of two-dimensional infinite group problems. Math. Oper. Res. 33, 140–166 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. In: Lodi, A., Panconesci, A., Rinaldi, G. (eds.) Proceedings 13th Conference on Integer Programming and Combinatorial Optimization, pp. 463–475. Springer (2007)

  16. Dey, S.S., Wolsey, L.A.: Two row mixed integer cuts via lifting. Technical Report 30, CORE DP, Université catholique de Louvain, Louvain-la-Neuve, Belgium (2008)

  17. Fischetti M., Saturni C.: Mixed integer cuts from cyclic groups. Math. Prog. 109, 27–53 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gomory R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gomory, R.E.: An algorithm for the mixed-integer problem. Technical Report RM-2597, Rand Report (1960)

  20. Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part I. Math. Prog. 3, 23–85 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part II. Math. Prog. 3, 359–389 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gomory R.E., Johnson E.L.: T-space and cutting planes. Math. Prog. 96, 341–375 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gomory R.E., Johnson E.L., Evans L.: Corner polyhedra and their connection with cutting planes. Math. Prog. 96, 321–339 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Johnson E.L.: On the group problem for mixed integer programming. Math. Prog. Study 2, 137–179 (1974)

    Google Scholar 

  25. Johnson E.L.: Characterization of facets for multiple right-hand choice linear programs. Math. Prog. Study 14, 112–142 (1981)

    MATH  Google Scholar 

  26. Lovász L.: Geometry of numbers and integer programming. In: Iri, M., Tanabe, K. (eds) Mathematical Programming: Recent Developments and Applications, Kluwer, Dordrecht (1989)

    Google Scholar 

  27. Miller L.A., Li Y., Richard J.-P.P.: New facets for finite and infinite group problems from approximate lifting. Naval Res. Logis. 55, 172–191 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    MATH  Google Scholar 

  30. Richard, J.-P.P., Li, Y., Miller, L.A.: Valid inequalities for MIPs and group polyhedra from approximate liftings. Math. Prog. 118, 253–277 (2009)

    Google Scholar 

  31. Rockafeller G.T.: Convex Analysis. Princeton University Press, New Jersey, NJ (1970)

    Google Scholar 

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Authors and Affiliations

  1. CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Santanu S. Dey

  2. CORE and INMA, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Laurence A. Wolsey

Authors
  1. Santanu S. Dey
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  2. Laurence A. Wolsey
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Correspondence to Santanu S. Dey.

Additional information

This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

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Dey, S.S., Wolsey, L.A. Two row mixed-integer cuts via lifting. Math. Program. 124, 143–174 (2010). https://doi.org/10.1007/s10107-010-0362-x

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  • DOI: https://doi.org/10.1007/s10107-010-0362-x

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