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Lifting, superadditivity, mixed integer rounding and single node flow sets revisited

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  • LOUVEAUX, Quentin
  • WOLSEY, Laurence

Abstract

In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0-1 single node flow set.

Suggested Citation

  • LOUVEAUX, Quentin & WOLSEY, Laurence, 2003. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," LIDAM Discussion Papers CORE 2003001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2003001
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp2003.html
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    Citations

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    Cited by:

    1. Lisa A. Miller & Yanjun Li & Jean‐Philippe P. Richard, 2008. "New inequalities for finite and infinite group problems from approximate lifting," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(2), pages 172-191, March.
    2. Agra, Agostinho & Constantino, Miguel Fragoso, 2016. "Lifted Euclidean inequalities for the integer single node flow set with upper bounds," European Journal of Operational Research, Elsevier, vol. 251(1), pages 53-63.
    3. Akbalik, A. & Pochet, Y., 2009. "Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs," European Journal of Operational Research, Elsevier, vol. 198(2), pages 412-434, October.
    4. Denis Bouyssou & Silvano Martello & Frank Plastria, 2010. "Eleven surveys in operations research: II," Annals of Operations Research, Springer, vol. 175(1), pages 3-8, March.
    5. Leo Liberti & Thierry Marchant & Silvano Martello, 2013. "Eleven surveys in operations research: III," Annals of Operations Research, Springer, vol. 204(1), pages 3-9, April.
    6. Alper Atamtürk, 2005. "Cover and Pack Inequalities for (Mixed) Integer Programming," Annals of Operations Research, Springer, vol. 139(1), pages 21-38, October.
    7. Leo Liberti & Thierry Marchant & Silvano Martello, 2016. "Twelve surveys in operations research," Annals of Operations Research, Springer, vol. 240(1), pages 3-11, May.
    8. Yves Crama & Michel Grabisch & Silvano Martello, 2018. "Surveys in operations research," Annals of Operations Research, Springer, vol. 271(1), pages 3-10, December.
    9. Yves Crama & Michel Grabisch & Silvano Martello, 2022. "Sixty-one surveys in operations research," Annals of Operations Research, Springer, vol. 314(1), pages 5-13, July.
    10. Avella, P. & Boccia, M. & Mattia, S. & Rossi, F., 2021. "Weak flow cover inequalities for the capacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 289(2), pages 485-494.
    11. Quentin Louveaux & Laurence Wolsey, 2007. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," Annals of Operations Research, Springer, vol. 153(1), pages 47-77, September.

    More about this item

    Keywords

    lifting; mixed integer rounding; single node flow sets;
    All these keywords.

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General

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