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Searching for optimal integer solutions to set partitioning problems using column generation

Author

Listed:
  • Bredström, David

    (Dept. of Mathematics, Linköping University)

  • Jörnsten, Kurt

    (Dept. of Finance and Management Science, Norwegian School of Economics and Business Administration)

  • Rönnqvist, Mikael

    (Dept. of Finance and Management Science, Norwegian School of Economics and Business Administration)

Abstract

We describe a new approach to produce integer feasible columns to a set partitioning problem directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP relaxation as quick as possible without any concern of the integer properties of the columns formed. In our approach we aim to generate the columns forming the optimal integer solution while simultaneously solving the LP relaxation. By this we can remove column generation in the branch and bound search. The basis is a subgradient technique applied to a Lagrangian dual formulation of the set partitioning problem extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate the optimal integer columns in a large set of well known test problems as compared to both standard and stabilized column generation and simultaneously keep the number of columns smaller than standard column generation.

Suggested Citation

  • Bredström, David & Jörnsten, Kurt & Rönnqvist, Mikael, 2007. "Searching for optimal integer solutions to set partitioning problems using column generation," Discussion Papers 2007/20, Norwegian School of Economics, Department of Business and Management Science.
    • Handle: RePEc:hhs:nhhfms:2007_020
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    File URL: http://hdl.handle.net/11250/227271
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    References listed on IDEAS

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    1. Karla L. Hoffman & Manfred Padberg, 1993. "Solving Airline Crew Scheduling Problems by Branch-and-Cut," Management Science, INFORMS, vol. 39(6), pages 657-682, June.
    2. Cynthia Barnhart & Ellis L. Johnson & George L. Nemhauser & Martin W. P. Savelsbergh & Pamela H. Vance, 1998. "Branch-and-Price: Column Generation for Solving Huge Integer Programs," Operations Research, INFORMS, vol. 46(3), pages 316-329, June.
    3. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Linear Programming; Branch and Bound tree; Lagrangian dual formulation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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