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Another pedagogy for mixed-integer Gomory

Author

Listed:
  • Jon Lee

    (University of Michigan)

  • Angelika Wiegele

    (Alpen-Adria-Universität Klagenfurt)

Abstract

We present a version of GMI (Gomory mixed-integer) cuts in a way so that they are derived with respect to a “dual form” mixed-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. This follows the general scheme of He and Lee, who did the case of Gomory pure-integer cuts. Our input mixed-integer problem is not in standard form, and so our cuts are derived rather differently from how they are normally derived. A convenient way to develop GMI cuts is from MIR (mixed-integer rounding) cuts, which are developed from 2-dimensional BMI (basic mixed-integer) cuts, which involve a nonnegative continuous variable and an integer variable. The non-negativity of the continuous variable is not the right tool for us, as our starting point (the “dual form” mixed-integer optimization problem) has no non-negativity. So we work out a different 2-dimensional starting point, a pair of somewhat arbitrary inequalities in one continuous and one integer variable. In the end, we follow the approach of He and Lee, getting now a finitely converging primal simplex column-generation algorithm for mixed-integer optimization problems.

Suggested Citation

  • Jon Lee & Angelika Wiegele, 2017. "Another pedagogy for mixed-integer Gomory," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(4), pages 455-466, December.
  • Handle: RePEc:spr:eurjco:v:5:y:2017:i:4:d:10.1007_s13675-017-0085-3
    DOI: 10.1007/s13675-017-0085-3
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    References listed on IDEAS

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    1. 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).
    2. M. L. Balinski, 1965. "Integer Programming: Methods, Uses, Computations," Management Science, INFORMS, vol. 12(3), pages 253-313, November.
    3. Hugues Marchand & Laurence A. Wolsey, 2001. "Aggregation and Mixed Integer Rounding to Solve MIPs," Operations Research, INFORMS, vol. 49(3), pages 363-371, June.
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    Cited by:

    1. Michele Conforti & Marianna Santis & Marco Summa & Francesco Rinaldi, 2021. "Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective," 4OR, Springer, vol. 19(4), pages 531-548, December.

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