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The minimum mean cycle-canceling algorithm for linear programs

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  • Gauthier, Jean Bertrand
  • Desrosiers, Jacques

Abstract

This paper presents the properties of the minimum mean cycle-canceling algorithm for solving linear programming models. Originally designed for solving network flow problems for which it runs in strongly polynomial time, most of its properties are preserved. This is at the price of adapting the fundamental decomposition theorem of a network flow solution together with various definitions: that of a cycle and the way to calculate its cost, the residual problem, and the improvement factor at the end of a phase. We also use the primal and dual necessary and sufficient optimality conditions stated on the residual problem for establishing the pricing step giving its name to the algorithm. It turns out that the successive solutions need not be basic, there are no degenerate pivots, and the improving directions are potentially interior in addition to those on edges. For solving an m×n linear program, it requires a pseudo-polynomial number O(nΔ) of so-called phases, where Δ depends on the number of rows and the coefficient matrix.

Suggested Citation

  • Gauthier, Jean Bertrand & Desrosiers, Jacques, 2022. "The minimum mean cycle-canceling algorithm for linear programs," European Journal of Operational Research, Elsevier, vol. 298(1), pages 36-44.
    • Handle: RePEc:eee:ejores:v:298:y:2022:i:1:p:36-44
    DOI: 10.1016/j.ejor.2021.09.022
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    References listed on IDEAS

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    1. Issmail Elhallaoui & Abdelmoutalib Metrane & Guy Desaulniers & François Soumis, 2011. "An Improved Primal Simplex Algorithm for Degenerate Linear Programs," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 569-577, November.
    2. A. Pessoa & R. Sadykov & E. Uchoa & F. Vanderbeck, 2018. "Automation and Combination of Linear-Programming Based Stabilization Techniques in Column Generation," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 339-360, May.
    3. Issmail Elhallaoui & Daniel Villeneuve & François Soumis & Guy Desaulniers, 2005. "Dynamic Aggregation of Set-Partitioning Constraints in Column Generation," Operations Research, INFORMS, vol. 53(4), pages 632-645, August.
    4. A. Charnes & W. W. Cooper, 1962. "Programming with linear fractional functionals," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 9(3‐4), pages 181-186, September.
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