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Criss-cross methods: A fresh view on pivot algorithms

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Abstract

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon.

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Authors and Affiliations

  1. Department of Mathematics, Swiss Federal Institute of Technology, Lausanne, Switzerland

    Komei Fukuda

  2. Institute for Operations Research, Swiss Federal Institute of Technology, Zürich, Switzerland

    Komei Fukuda

  3. Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Tamás Terlaky

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  1. Komei Fukuda
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  2. Tamás Terlaky
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Fukuda, K., Terlaky, T. Criss-cross methods: A fresh view on pivot algorithms. Mathematical Programming 79, 369–395 (1997). https://doi.org/10.1007/BF02614325

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