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In this paper we analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem ofn variables withm constraints. Smale established that for every number of constraintsm, there is a constantc(m) such that the number of pivot steps of the self-dual algorithm,?(m, n), is less thanc(m)(lnn)m(m+1). We improve upon this estimate by showing that?(m, n) is bounded by a function ofm only. The symmetry of the function inm andn implies that?(m, n) is in fact bounded by a function of the smaller ofm andn.

Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm