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Aggregation and Mixed Integer Rounding to Solve MIPs

Author

Listed:
  • Hugues Marchand

    (Department of Operational Research, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom)

  • Laurence A. Wolsey

    (Center for Operations Research and Econometrics, Université Catholique de Louvain, Voie du Roman pays, 34, Louvain-la-Neuve B-1348, Belgium)

Abstract

In this paper, we discuss the use of mixed integer rounding (MIR) inequalities to solve mixed integer programs. MIR inequalities are essentially Gomory mixed integer cuts. However, as we wish to use problem structure, we insist that MIR inequalities be generated from constraints or simple aggregations of constraints of the original problem. This idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as MIR inequalities.Here we present and test a separation routine for such MIR inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint, complementation of variables, and finally the generation of an MIR inequality. Inserted in a branch-and-cut system, the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems.

Suggested Citation

  • Hugues Marchand & Laurence A. Wolsey, 2001. "Aggregation and Mixed Integer Rounding to Solve MIPs," Operations Research, INFORMS, vol. 49(3), pages 363-371, June.
  • Handle: RePEc:inm:oropre:v:49:y:2001:i:3:p:363-371
    DOI: 10.1287/opre.49.3.363.11211
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    References listed on IDEAS

    as
    1. Pochet, Y. & Wolsey, L. A., 1995. "Integer knapsack and flow covers with divisible coefficients: polyhedra, optimization and separation," LIDAM Reprints CORE 1149, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. 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).
    3. Daniel Bienstock & Oktay Günlük, 1996. "Capacitated Network Design---Polyhedral Structure and Computation," INFORMS Journal on Computing, INFORMS, vol. 8(3), pages 243-259, August.
    4. Egon Balas & Sebastián Ceria & Gérard Cornuéjols, 1996. "Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework," Management Science, INFORMS, vol. 42(9), pages 1229-1246, September.
    5. 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).
    6. BARANY, Imre & VAN ROY, Tony J. & WOLSEY, Laurence A., 1984. "Strong formulations for multi-item capacitated lot sizing," LIDAM Reprints CORE 590, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. 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.
    8. Imre Barany & Tony J. Van Roy & Laurence A. Wolsey, 1984. "Strong Formulations for Multi-Item Capacitated Lot Sizing," Management Science, INFORMS, vol. 30(10), pages 1255-1261, October.
    9. Padberg, M.W. & Van Roy, T.J. & Wolsey, L.A., 1985. "Valid linear inequalities for fixed charge problems," LIDAM Reprints CORE 656, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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