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Intersection Cuts and Corner Polyhedra

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Book cover Integer Programming

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

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Abstract

In this chapter, we present two classical points of view for approximating a mixed integer linear set: Gomory’s corner polyhedron and Balas’ intersection cuts. It turns out that they are equivalent: the nontrivial valid inequalities for the corner polyhedron are exactly the intersection cuts. Within this framework, we stress two ideas: the best possible intersection cuts are generated from maximal lattice-free convex sets, and formulas for these cuts can be interpreted using the so-called infinite relaxation.

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Bibliography

  1. K. Andersen, Q. Louveaux, R. Weismantel, L.A. Wolsey, Inequalities from two rows of a simplex tableau, in Proceedings of IPCO XII, Ithaca, NY. Lecture Notes in Computer Science, vol. 4513 (2007), pp. 1–15

    Article  MathSciNet  Google Scholar 

  2. G. Averkov, On maximal S-free sets and the Helly number for the family of S-convex sets. SIAM J. Discrete Math. 27(3), 1610–1624 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Averkov, A. Basu, On the unique lifting property, IPCO 2014, Bonn, Germany, Lecture Notes in Computer Science, 8494, 76–87 (2014)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Balas, Integer programming and convex analysis: intersection cuts from outer polars. Math. Program. 2 330–382 (1972)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, 2002)

    Google Scholar 

  8. A. Basu, M. Campelo, M. Conforti, G. Cornuéjols, G. Zambelli, On lifting integer variables in minimal inequalities. Math. Program. A 141, 561–576 (2013)

    Article  MATH  Google Scholar 

  9. A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discrete Math. 24, 158–168 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Basu, R. Hildebrand, M. Köppe, M. Molinaro, A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation. SIAM J. Optim. 23(2), 1021–1040 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Basu, R. Hildebrand, M. Köppe, Equivariant perturbation in Gomory and Johnson infinite group problem III. Foundations for the k-dimensional case with applications to the case k = 2. www.optimization-online.org (2014)

  13. D.E. Bell, A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)

    MATH  Google Scholar 

  14. V. Borozan, G. Cornuéjols, Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Conforti, G. Cornuéjols, A. Daniilidis, C. Lemaréchal, J. Malick, Cut-generating functions and S-free sets, Math. Oper. Res. http://dx.doi.org/10.1287/moor.2014.0670

  16. M. Conforti, G. Cornuéjols, G. Zambelli, A geometric perspective on lifting. Oper. Res. 59, 569–577 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Conforti, G. Cornuéjols, G. Zambelli, Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38, 153–155 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Conforti, G. Cornuéjols, G. Zambelli, Corner polyhedron and intersection cuts. Surv. Oper. Res. Manag. Sci. 16, 105–120 (2011)

    Google Scholar 

  19. S. Dash, S.S. Dey, O. Günlük, Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra. Math. Program. 135, 221–254 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Del Pia, R. Weismantel, Relaxations of mixed integer sets from lattice-free polyhedra. 4OR 10, 221–244 (2012)

    Google Scholar 

  21. S.S. Dey, Q. Louveaux, Split rank of triangle and quadrilateral inequalities. Math. Oper. Res. 36, 432–461 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. S. S. Dey, D.A. Morán, On maximal S-free convex sets. SIAM J. Discrete Math. 25(1), 379–393 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. S.S. Dey, J.-P.P. Richard, Y. Li, L.A. Miller, On the extreme inequalities of infinite group problems. Math. Program. A 121, 145–170 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. S.S. Dey, L.A. Wolsey, Lifting Integer Variables in Minimal Inequalities Corresponding to Lattice-Free Triangles, IPCO 2008, Bertinoro, Italy. Lecture Notes in Computer Science, Springer, vol. 5035 (2008), pp. 463–475

    Article  MathSciNet  Google Scholar 

  25. S.S. Dey, L.A. Wolsey, Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20, 2890–2912 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. J.-P. Doignon, Convexity in cristallographical lattices. J. Geom. 3, 71–85 (1973)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. R.E. Gomory, E.L. Johnson, Some continuous functions related to corner polyhedra I. Math. Program. 3, 23–85 (1972)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. J.-B. Hiriart-Urruty, C. Lemaréchal. Fundamentals of Convex Analysis (Springer, New York, 2001)

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. L. Lovász, Geometry of numbers and integer programming, in Mathematical Programming: Recent Developments and Applications, ed. by M. Iri, K. Tanabe (Kluwer, Dordrecht, 1989), pp. 177–201

    Google Scholar 

  34. J.-P.P. Richard, S.S. Dey (2010). The group-theoretic approach in mixed integer programming, in 50 Years of Integer Programming 1958–2008, ed. by M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (Springer, New York, 2010), pp. 727–801

    Google Scholar 

  35. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1969)

    Google Scholar 

  36. H.E. Scarf, An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. USA 74, 3637–3641 (1977)

    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). Intersection Cuts and Corner Polyhedra. In: Integer Programming. Graduate Texts in Mathematics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-319-11008-0_6

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