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Split and Gomory Inequalities

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Integer Programming

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 271))

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Abstract

Chapter 4 dealt with perfect formulations. What can one do when one is handed a formulation that is not perfect? A possible option is to strengthen the formulation in an attempt to make it closer to being perfect. One of the most successful strengthening techniques in practice is the addition of Gomory’s mixed integer cuts. These inequalities have a geometric interpretation, in the context of Balas’ disjunctive programming. They are known as split inequalities in this context, and they are the topic of interest in this chapter. They are also related to the so-called mixed integer rounding inequalities. We show that the convex set defined by intersecting all split inequalities is a polyhedron. For pure integer problems and mixed 0,1 problems, iterating this process a finite number of times produces a perfect formulation. We study Chvátal inequalities and lift-and-project inequalities, which are important special cases of split inequalities. Finally, we discuss cutting planes algorithms based on Gomory inequalities and lift-and-project inequalities, and provide convergence results.

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Bibliography

  1. K. Andersen, G. Cornuéjols, Y. Li, Split closure and intersection cuts. Math. Program. A 102, 457–493 (2005)

    Article  MATH  Google Scholar 

  2. E. Balas, Disjunctive programming: properties of the convex hull of feasible points, GSIA Management Science Research Report MSRR 348, Carnegie Mellon University (1974); Published as invited paper in Discrete Appl. Math. 89, 1–44 (1998)

    Google Scholar 

  3. E. Balas, A modified lift-and-project procedure. Math. Program. 79, 19–31 (1997)

    MATH  MathSciNet  Google Scholar 

  4. E. Balas, P. Bonami, 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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. E. Balas, S. Ceria, G. Cornuéjols, R.N. Natraj, Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Balas, A. Saxena, Optimizing over the split closure. Math. Program. 113, 219–240 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. R.E. Bixby, S. Ceria, C.M. McZeal, M.W.P. Savelsbergh, An updated mixed integer programming library: MIPLIB 3.0. Optima 58, 12–15 (1998)

    Google Scholar 

  11. R.E. Bixby, M. Fenelon, Z. Gu, E. Rothberg, R. Wunderling, Mixed integer programming: a progress report, in The Sharpest Cut: The Impact of Manfred Padberg and His Work, ed. by M. Grötschel. MPS/SIAM Series in Optimization (2004), pp. 309–326

    Google Scholar 

  12. P. Bonami, On optimizing over lift-and-project closures. Math. Program. Comput. 4, 151–179 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Bonami, M. Conforti, G. Cornuéjols, M. Molinaro, G. Zambelli, Cutting planes from two-term disjunctions. Oper. Res. Lett. 41, 442–444 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Bonami, G. Cornuéjols, S. Dash, M. Fischetti, A. Lodi, Projected Chvátal-Gomory cuts for mixed integer linear programs. Math. Program. 113, 241–257 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Bonami, F. Margot, Cut generation through binarization, IPCO 2014, eds. by J. Lee, J. Vygen. LNCS, vol 8494 (2014) pp. 174–185

    Google Scholar 

  16. A. Caprara, M. Fischetti, \(\{0, \frac{1} {2}\}\) Chvátal–Gomory cuts. Math. Program. 74, 221–235 (1996)

    Google Scholar 

  17. A. Caprara, A.N. Letchford, On the separation of split cuts and related inequalities. Math. Program. B 94, 279–294 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. V. Chvátal, W. Cook, M. Hartmann, On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114/115, 455–499 (1989)

    Google Scholar 

  20. M. Conforti, L.A. Wolsey, G. Zambelli, Split, MIR and Gomory inequalities (2012 submitted)

    Google Scholar 

  21. W.J. Cook, S. Dash, R. Fukasawa, M. Goycoolea, Numerically accurate Gomory mixed-integer cuts. INFORMS J. Comput. 21, 641–649 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Cornuéjols, Y. Li, On the rank of mixed 0,1 polyhedra. Math. Program. A 91, 391–397 (2002)

    Article  MATH  Google Scholar 

  24. G. Cornuéjols, Y. Li, A connection between cutting plane theory and the geometry of numbers. Math. Program. A 93, 123–127 (2002)

    Article  MATH  Google Scholar 

  25. S. Dash, O. Günlük, A. Lodi, in On the MIR Closure of Polyhedra, IPCO 2007, ed. by M. Fischetti, D.P. Williamson. LNCS, Springer vol. 4513 (2007), pp. 337–351

    Google Scholar 

  26. A. Del Pia, R. Weismantel, On convergence in mixed integer programming. Math. Program. 135, 397–412 (2012)

    Google Scholar 

  27. F. Eisenbrand, On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. F. Eisenbrand, A.S. Schulz, Bounds on the Chvátal rank of polytopes in the 0/1 cube. Combinatorica 23, 245–261 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Fischetti, A. Lodi, Optimizing over the first Chvátal closure. Math. Program. 110, 3–20 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. R.S. Garfinkel, G. Nemhauser, Integer Programming (Wiley, New York, 1972)

    MATH  Google Scholar 

  32. R.E. Gomory, Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  33. R.E. Gomory, An algorithm for the mixed integer problem. Tech. Report RM-2597 (The Rand Corporation, 1960)

    Google Scholar 

  34. R.E. Gomory, An algorithm for integer solutions to linear programs, in Recent Advances in Mathematical Programming, ed. by R.L. Graves, P. Wolfe (McGraw-Hill, New York, 1963), pp. 269–302

    Google Scholar 

  35. M. Köppe, Q. Louveaux, R. Weismantel, Intermediate integer programming representations using value disjunctions. Discrete Optim. 5, 293–313 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. H. Marchand, L.A. Wolsey, Aggregation and mmixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  38. J.H. Owen, S. Mehrotra, A disjunctive cutting plane procedure for general mixed-integer linear programs. Math. Program. A 89, 437–448 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  39. J.H. Owen, S. Mehrotra, On the value of binary expansions for general mixed-integer linear programs. Oper. Res. 50, 810–819 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  40. T. Rothvoß, L. Sanitá, 0 − 1 polytopes with quadratic Chvátal rank, in Proceedings of the 16th IPCO Conference. Lecture Notes in Computer Science, vol. 7801 (Springer, New York, 2013)

    Google Scholar 

  41. J.-S. Roy, Reformulation of bounded integer variables into binary variables to generate cuts. Algorithmic Oper. Res. 2, 810–819 (2007)

    Google Scholar 

  42. A. Schrijver, On cutting planes. Ann. Discrete Math. 9, 291–296 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  43. A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986)

    MATH  Google Scholar 

  44. H. Sherali, W. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Chap. 4 (Kluwer Academic Publishers, Norwell, 1999)

    Google Scholar 

  45. J.P. Vielma, A constructive characterization of the split closure of a mixed integer linear program. Oper. Res. Lett. 35, 29–35 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Padova, Padova, Italy

    Michele Conforti

  2. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    Gérard Cornuéjols

  3. Department of Management, London School of Economics and Political Science, London, UK

    Giacomo Zambelli

Authors
  1. Michele Conforti
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  2. Gérard Cornuéjols
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  3. Giacomo Zambelli
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Conforti, M., Cornuéjols, G., Zambelli, G. (2014). Split and Gomory Inequalities. In: Integer Programming. Graduate Texts in Mathematics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-319-11008-0_5

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