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Relaxations of mixed integer sets from lattice-free polyhedra

Author

Listed:
  • Alberto Del Pia

    (University of Wisconsin-Madison)

  • Robert Weismantel

    (ETH Zürich)

Abstract

This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitely.

Suggested Citation

  • Alberto Del Pia & Robert Weismantel, 2016. "Relaxations of mixed integer sets from lattice-free polyhedra," Annals of Operations Research, Springer, vol. 240(1), pages 95-117, May.
    • Handle: RePEc:spr:annopr:v:240:y:2016:i:1:d:10.1007_s10479-015-2024-0
    DOI: 10.1007/s10479-015-2024-0
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    References listed on IDEAS

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    1. Sanjeeb Dash & Oktay Günlük & Juan Pablo Vielma, 2014. "Computational Experiments with Cross and Crooked Cross Cuts," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 780-797, November.
    2. Amitabh Basu & Gérard Cornuéjols & Matthias Köppe, 2012. "Unique Minimal Liftings for Simplicial Polytopes," Mathematics of Operations Research, INFORMS, vol. 37(2), pages 346-355, May.
    3. Gennadiy Averkov & Christian Wagner & Robert Weismantel, 2011. "Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 721-742, November.
    4. Alberto Del Pia, 2012. "On the Rank of Disjunctive Cuts," Mathematics of Operations Research, INFORMS, vol. 37(2), pages 372-378, May.
    5. Egon Balas, 1971. "Intersection Cuts—A New Type of Cutting Planes for Integer Programming," Operations Research, INFORMS, vol. 19(1), pages 19-39, February.
    6. Schrijver, A, 1980. "On Cutting Planes," University of Amsterdam, Actuarial Science and Econometrics Archive 293054, University of Amsterdam, Faculty of Economics and Business.
    7. Benjamin Nill & Günter M. Ziegler, 2011. "Projecting Lattice Polytopes Without Interior Lattice Points," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 462-467, August.
    8. SALINETTI, Gabriella & WETS, Roger J.-B., 1979. "On the convergence of sequences of convex sets in finite dimensions," LIDAM Reprints CORE 352, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. MARCHAND, Hugues & WOLSEY, Laurence A., 2001. "Aggregation and mixed integer rounding to solve mips," LIDAM Reprints CORE 1513, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Kent Andersen & Quentin Louveaux & Robert Weismantel, 2010. "An Analysis of Mixed Integer Linear Sets Based on Lattice Point Free Convex Sets," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 233-256, February.
    11. Hugues Marchand & Laurence A. Wolsey, 2001. "Aggregation and Mixed Integer Rounding to Solve MIPs," Operations Research, INFORMS, vol. 49(3), pages 363-371, June.
    12. Michele Conforti & Gérard Cornuéjols & Giacomo Zambelli, 2011. "A Geometric Perspective on Lifting," Operations Research, INFORMS, vol. 59(3), pages 569-577, June.
    13. Amitabh Basu & Michele Conforti & Gérard Cornuéjols & Giacomo Zambelli, 2010. "Maximal Lattice-Free Convex Sets in Linear Subspaces," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 704-720, August.
    14. NEMHAUSER, George L. & WOLSEY, Laurence A., 1990. "A recursive procedure to generate all cuts for 0-1 mixed integer programs," LIDAM Reprints CORE 894, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. Amitabh Basu & Gérard Cornuéjols & François Margot, 2012. "Intersection Cuts with Infinite Split Rank," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 21-40, February.
    16. DEY, Santanu S. & WOLSEY, Laurence A., 2010. "Two row mixed-integer cuts via lifting," LIDAM Reprints CORE 2254, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. Santanu S. Dey & Quentin Louveaux, 2011. "Split Rank of Triangle and Quadrilateral Inequalities," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 432-461, August.
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