The Simplex Method: A Probabilistic Analysis by Karl Heinz Borgwardt (English) P

Description: The Simplex Method by Karl Heinz Borgwardt While the "worst-case analysis" of some variants of the method shows that this is not a "good" algorithm in the usual sense of complexity theory, it seems to be useful to apply other criteria for a judgement concerning the quality of the algorithm. FORMAT Paperback LANGUAGE English CONDITION Brand New Publisher Description For more than 35 years now, George B. Dantzigs Simplex-Method has been the most efficient mathematical tool for solving linear programming problems. It is proba bly that mathematical algorithm for which the most computation time on computers is spent. This fact explains the great interest of experts and of the public to understand the method and its efficiency. But there are linear programming problems which will not be solved by a given variant of the Simplex-Method in an acceptable time. The discrepancy between this (negative) theoretical result and the good practical behaviour of the method has caused a great fascination for many years. While the "worst-case analysis" of some variants of the method shows that this is not a "good" algorithm in the usual sense of complexity theory, it seems to be useful to apply other criteria for a judgement concerning the quality of the algorithm. One of these criteria is the average computation time, which amounts to an anal ysis of the average number of elementary arithmetic computations and of the number of pivot steps. A rigid analysis of the average behaviour may be very helpful for the decision which algorithm and which variant shall be used in practical applications. The subject and purpose of this book is to explain the great efficiency in prac tice by assuming certain distributions on the "real-world" -problems. Other stochastic models are realistic as well and so this analysis should be considered as one of many possibilities. Notes Springer Book Archives Table of Contents 0 Introduction.- Formulation of the problem and basic notation.- 1 The problem.- A Historical Overview.- 2 The gap between worst case and practical experience.- 3 Alternative algorithms.- 4 Results of stochastic geometry.- 5 The results of the author.- 6 The work of Smale.- 7 The paper of Haimovich.- 8 Quadratic expected number of steps for sign-invariance model.- Discussion of different stochastic models.- 9 What is the "Real World Model"?.- Outline of Chapters 1–5.- 10 The basic ideas and the methods of this book.- 11 The results of this book.- 12 Conclusion and conjectures.- 1 The Shadow-Vertex Algorithm.- 1 Primal interpretation.- 2 Dual interpretation.- 3 Numerical realization of the algorithm.- 4 The algorithm for Phase I.- 2 The Average Number of Pivot Steps.- 1 The probability space.- 2 An integral formula for the expected number of S.- 3 A transformation of coordinates.- 4 Generalizations.- 3 The Polynomiality of the Expected Number of Steps.- 1 Comparison of two integrals.- 2 An application of Cavalieris Principle.- 3 The influence of the distribution.- 4 Evaluation of the quotient.- 5 The average number of steps in our complete Simplex-Method.- 4 Asymptotic Results.- 1 An asymptotic upper bound in integral form.- 2 Asymptotic results for certain classes of distributions.- 3 Special distributions with bounded support.- 4 Asymptotic bounds under uniform distributions.- 5 Asymptotic bounds under Gaussian distribution.- 5 Problems with Nonnegativity Constraints.- 1 The geometry.- 2 The complete solution method.- 3 A simplification of the boundary-condition.- 4 Explicit formulation of the intersection-condition.- 5 Componentwise sign-independence and the intersection condition.- 6 The average number of pivot steps.- 6 Appendix.- 1 Gammafunction andBetafunction.- 2 Unit ball and unit sphere.- 3 Estimations under variation of the weights.- References. Promotional Springer Book Archives Long Description For more than 35 years now, George B. Dantzigs Simplex-Method has been the most efficient mathematical tool for solving linear programming problems. It is proba Details ISBN3540170960 Publisher Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Series Algorithms and Combinatorics ISBN-10 3540170960 ISBN-13 9783540170969 Format Paperback Imprint Springer-Verlag Berlin and Heidelberg GmbH & Co. K Subtitle A Probabilistic Analysis Place of Publication Berlin Country of Publication Germany DEWEY 004.0151972 Year 1986 Publication Date 1986-11-01 Language English Media Book Series Number 1 Short Title SIMPLEX METHOD SOFTCOVER REPRI Author Karl Heinz Borgwardt Pages 270 Illustrations 3 Illustrations, black and white; XII, 270 p. 3 illus. DOI 10.1007/b33342;10.1007/978-3-642-61578-8 Edition Description Softcover reprint of the original 1st ed. 1987 Audience Undergraduate We've got this At The Nile, if you're looking for it, we've got it. With fast shipping, low prices, friendly service and well over a million items - you're bound to find what you want, at a price you'll love! TheNile_Item_ID:96287346;

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