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An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming

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Abstract

In recent years many advances have been made in solution techniques for specially structured 0-1 integer programming problems. In contrast, very little progress has been made on solving general (mixed integer) problems. This, of course, is not true when viewed from the theoretical side: Lenstra (1981) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of Lovasz and Scarf (1988).This method allows us to avoid the ellipsoidal approximations required in Lenstra's algorithm. We report on the solution of a number of small (but difficult) examples, up to 100 integer variables. Our computer code uses the linear programming optimizer CPlex as a subroutine to solve the linear programming problems that arise.

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  • William Cook & Thomas Rutherford & Herbert E. Scarf & David F. Shallcross, 1991. "An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming," Cowles Foundation Discussion Papers 990, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:990
    Note: CFP 906.
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d09/d0990.pdf
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    1. Herbert E. Scarf & Laszlo Lovasz, 1990. "The Generalized Basis Reduction Algorithm," Cowles Foundation Discussion Papers 946, Cowles Foundation for Research in Economics, Yale University.
    2. Ravi Kannan, 1987. "Minkowski's Convex Body Theorem and Integer Programming," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 415-440, August.
    3. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    4. Herbert E. Scarf & R. Kannan & Laszlo Lovasz, 1988. "The Shapes of Polyhedra," Cowles Foundation Discussion Papers 883, Cowles Foundation for Research in Economics, Yale University.
    5. VAN ROY, Tony J. & WOLSEY, Laurence A., 1987. "Solving mixed integer programming problems using automatic reformulation," LIDAM Reprints CORE 782, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Tony J. Van Roy & Laurence A. Wolsey, 1987. "Solving Mixed Integer Programming Problems Using Automatic Reformulation," Operations Research, INFORMS, vol. 35(1), pages 45-57, February.
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    More about this item

    Keywords

    Linear programming; mixed integer problems;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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