A lower bound on the average number of Pivot-steps for solving linear programs Valid for all variants of the Simplex-Algorithm

Abstract.

In this paper we derive a lower bound on the average complexity of the Simplex-Method as a solution-process for linear programs (LP) of the type:

We assume these problems to be randomly generated according to the Rotation-Symmetry-Model:

*Let a ₁,…,a _m, v be distributed independently, identically and symmetrically under rotations on ℝⁿ\{0}.

We concentrate on distributions over ℝⁿ with bounded support and we do our calculations only for a subfamily of such distributions, which provides computability and is representative for the whole set of these distributions.

 The Simplex-Method employs two phases to solve such an LP. In Phase I it determines a vertex x ₀ of the feasible region – if there is any. In Phase II it starts at x ₀ to generate a sequence of vertices x ₀,… ,x _s such that successive vertices are adjacent and that the objective v ^T x _i increases. The sequence ends at a vertex x _s which is either the optimal vertex or a vertex exhibiting the information that no optimal vertex can exist. The precise rule for choosing the successor-vertex in the sequence determines a variant of the Simplex-Algorithm.

 We can show for every variant, that the expected number of steps s ^var for a variant, when m inequalities and n variables are present, satisfies

This result holds, if the selection of x ₀ in Phase I has been done independently of the objective v.