On the Variance of the Number of Pivot Steps Required by the Simplex Algorithm

  • Karl-Heinz Küfer
Conference paper


Introduction: Despite their very good empirical performance the known variants of the simplex algorithm require exponentially many pivot steps in terms of the problem dimensions of the given linear programming problem (LPP) in worst-case situtation. The first to explain the large gap between practical experience and the disappointing worstcase was Borgwardt (1982a,b), who could prove polynomiality on the average for a certain variant of the algorithm—the “Schatteneckenalgorithmus (shadow vertex algorithm)”—using a stochastic problem simulation.


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    Borgwardt, K. H. (1982) Some Distribution-Independent Results About tbe Asymptotic Order of the Average Number of Pivot Steps of tbe Simplex Method. Math. of O R. 7: 441–462.CrossRefGoogle Scholar
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    Borgwardt, K. H. (1982) The Average Number of Pivot Steps Required by Tbe Simplex Method is Polynomial. Z. f. O. R.: 157–177.Google Scholar
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    Küfer, K. H. (1992) Asymptotische Varianzanalysen in der stochastischen Polyedertheorie. Dissertation, Universität Kaiserslautern.Google Scholar
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    Küfer, K. H. (1992) A Simple Integral Representation for the Second Moments of Additive Random Variables on Stochastic Polyhedra. To appear, Preprint Universität Kaiserslautern.Google Scholar
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    Küfer, K. H. (1992) On tbe Variance of tbe Number of Pivot Steps Required by the Simplex Algorithm. To appear, Preprint Universität Kaiserslautern Google Scholar
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    Shamir, R. (1987) The Efficiency of tbe Simplex Method: A Survey Management Science 33: 241–262.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Karl-Heinz Küfer
    • 1
  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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