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A Geometric Perspective on Lifting

Author

Listed:
  • Michele Conforti

    (Dipartimento di Matematica Pura ed Applicata, Università di Padova, 35121 Padova, Italy)

  • Gérard Cornuéjols

    (Tepper School of Business, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

  • Giacomo Zambelli

    (London School of Economics and Political Sciences, London WC2A 2AE, United Kingdom)

Abstract

Recently it has been shown that minimal inequalities for a continuous relaxation of mixed-integer linear programs are associated with maximal lattice-free convex sets. In this paper, we show how to lift these inequalities for integral nonbasic variables by considering maximal lattice-free convex sets in a higher dimensional space. We apply this approach to several examples. In particular, we identify cases in which the lifting is unique.

Suggested Citation

  • Michele Conforti & Gérard Cornuéjols & Giacomo Zambelli, 2011. "A Geometric Perspective on Lifting," Operations Research, INFORMS, vol. 59(3), pages 569-577, June.
    • Handle: RePEc:inm:oropre:v:59:y:2011:i:3:p:569-577
    DOI: 10.1287/opre.1110.0916
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    References listed on IDEAS

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    1. L. A. Wolsey, 1977. "Valid Inequalities and Superadditivity for 0--1 Integer Programs," Mathematics of Operations Research, INFORMS, vol. 2(1), pages 66-77, February.
    2. Valentin Borozan & Gérard Cornuéjols, 2009. "Minimal Valid Inequalities for Integer Constraints," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 538-546, August.
    3. Balas, Egon & Jeroslow, Robert G., 1980. "Strengthening cuts for mixed integer programs," European Journal of Operational Research, Elsevier, vol. 4(4), pages 224-234, April.
    4. Zonghao Gu & George L. Nemhauser & Martin W.P. Savelsbergh, 2000. "Sequence Independent Lifting in Mixed Integer Programming," Journal of Combinatorial Optimization, Springer, vol. 4(1), pages 109-129, March.
    5. DEY, Santanu S. & WOLSEY, Laurence A., 2010. "Constrained infinite group relaxations of MIPS," LIDAM Reprints CORE 2314, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. WOLSEY, Laurence A., 1977. "Valid inequalities and superadditivity for 0-1 integer programs," LIDAM Reprints CORE 328, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Santanu S. Dey & Andrea Lodi & Andrea Tramontani & Laurence A. Wolsey, 2014. "On the Practical Strength of Two-Row Tableau Cuts," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 222-237, May.
    4. 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.
    5. Sercan Yıldız & Gérard Cornuéjols, 2016. "Cut-Generating Functions for Integer Variables," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1381-1403, November.

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    programming/integer/cutting plane;

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