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

Abstract

Leta ₁ ...,a _m be i.i.d. points uniformly on the unit sphere in ℝⁿ,m ≥n ≥ 3, and letX:= {xε ℝⁿ|a ^T_i x≤1} be the random polyhedron generated bya ₁, ...,a _m . Furthermore, for linearly independent vectorsu, ū in ℝⁿ, 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 := ¹_in4 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.