Abstract
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,ρ(m, n), is less thanc(m)(lnn)m(m+1). We improve upon this estimate by showing thatρ(m, n) is bounded by a function ofm only. The symmetry of the function inm andn implies thatρ(m, n) is in fact bounded by a function of the smaller ofm andn.
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Parts of this research were done while the author was visiting Stanford University, XEROX- PARC, Carnegie-Mellon University and Northwestern University and was supported in part by the National Science Foundation under Grants MCS-8300984, ECS-8218181 and ECS-8121741.
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Megiddo, N. Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm. Mathematical Programming 35, 140–172 (1986). https://doi.org/10.1007/BF01580645
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DOI: https://doi.org/10.1007/BF01580645