Mathematical Methods of Operations Research

, Volume 44, Issue 2, pp 147–170 | Cite as

An improved asymptotic analysis of the expected number of pivot steps required by the simplex algorithm

  • Karl -Heinz Küfer
Article
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Revised:

Abstract

Leta1 ...,a m be i.i.d. points uniformly on the unit sphere in ℝ n ,m ≥n ≥ 3, and letX:= {xε ℝ n |a i T x≤1} be the random polyhedron generated bya1, ...,a m . Furthermore, for linearly independent vectorsu, ū in ℝ n , letS u ,ū (X) be the number of shadow vertices ofX inspan(u,ū). The paper provides an asymptotic expansion of the expectation value¯S n,m := in4 1 E(S u,ū ) for fixedn andm→ ∞.¯S n,m equals the expected number of pivot steps that the shadow vertex algorithm — a parametric variant of the simplex algorithm — requires in order to solve linear programming problems of type max u T ,xεX, if the algorithm will be started with anX-vertex solving the problem max ū T ,x ε X. Our analysis is closely related to Borgwardt's probabilistic analysis of the simplex algorithm. We obtain a refined asymptotic analysis of the expected number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data.

Key words

Linear programming simplex algorithm probabilistic analysis asymptotic expansion convex hull stochastic geometry 
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Copyright information

© Physica-Verlag 1996

Authors and Affiliations

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

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