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

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  • Quentin Louveaux
  • Laurence Wolsey

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. Copyright Springer Science+Business Media, LLC 2007

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:153:y:2007:i:1:p:47-77:10.1007/s10479-007-0171-7
    DOI: 10.1007/s10479-007-0171-7
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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. Laurence A. Wolsey, 1976. "Technical Note—Facets and Strong Valid Inequalities for Integer Programs," Operations Research, INFORMS, vol. 24(2), pages 367-372, April.
    3. LOUVEAUX, Quentin & WOLSEY, Laurence A., 2003. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," LIDAM Reprints CORE 1659, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. MARCHAND, Hugues & WOLSEY, Laurence A., 1999. "The 0-1 Knapsack problem with a single continuous variable," LIDAM Reprints CORE 1390, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. VAN ROY, Tony J. & WOLSEY, Laurence A., 1985. "Valid inequalities and separation for uncapacitated fixed charge networks," LIDAM Reprints CORE 671, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Hugues Marchand & Laurence A. Wolsey, 2001. "Aggregation and Mixed Integer Rounding to Solve MIPs," Operations Research, INFORMS, vol. 49(3), pages 363-371, June.
    7. 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).
    8. M. W. Padberg & T. J. Van Roy & L. A. Wolsey, 1985. "Valid Linear Inequalities for Fixed Charge Problems," Operations Research, INFORMS, vol. 33(4), pages 842-861, August.
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    10. WOLSEY, Laurence A., 1989. "Submodularity and valid inequalities in capacitated fixed charge networks," LIDAM Reprints CORE 842, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. 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).
    12. 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).
    13. CERIA, Sebastian & CORDIER, Cécile & MARCHAND, Hugues & WOLSEY, Laurence A., 1998. "Cutting planes for integer programs with general integer variables," LIDAM Reprints CORE 1334, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    15. VAN ROY, Tony J. & WOLSEY, Laurence A., 1986. "Valid inequalities for mixed 0-1 programs," LIDAM Reprints CORE 697, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    4. Detienne, Boris, 2014. "A mixed integer linear programming approach to minimize the number of late jobs with and without machine availability constraints," European Journal of Operational Research, Elsevier, vol. 235(3), pages 540-552.

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