Linear Programming

A Modern Integrated Analysis

AvRomesh Saigal

Inbunden, Engelska, 1995

Del i serien International Series in Operations Research & Management Science

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Beskrivning

In this study, both boundary (simplex) and interior point methods are derived from the complementary slackness theorem, and the duality theorem is derived from Farkas's Lemma, which is proved as a convex separation theorem. The tedium of the simplex method is thus avoided. A proof of Kantorovich's Theorem is offered, related to the convergence of Newton's method. Of the boundary methods, the book presents the (revised) primal and the dual simplex methods. A discussion is given of the primal, dual and primal-dual affine scaling methods. In addition, the proof of the convergence under degeneracy, a bounded variable variant, and a super-linearly convergent variant of the primal affine scaling method are covered in one chapter. Polynomial barrier or path-following homotopy methods, and the projective transformation method are also covered in the interior point chapter. Besides the popular sparse Cholesky factorization and the conjugate gradient method, new methods are presented in a separate chapter on implementation.

Produktinformation

Utgivningsdatum:1995-11-30
Mått:155 x 235 x 21 mm
Vikt:680 g
Format:Inbunden
Språk:Engelska
Serie:International Series in Operations Research & Management Science
Antal sidor:342
Upplaga:1995
Förlag:Kluwer Academic Publishers
ISBN:9780792396222

Utforska kategorier

Optimering inom Naturvetenskap och teknik
Företagsekonomi inom Ekonomi och Ledarskap

Recensioner i media

'I recommend this book to anyone desiring a deep understanding of the simplex method, interior-point methods, and the connections between them.' Interfaces, 27:2 (1997) The book is clearly written. ... It is highly recommended to anybody wishing to get a clear insight in the field and in the role that duality plays not only from a theoretical point of view but also in connection with algorithms.' Optimization, 40 (1997)

Innehållsförteckning

1 Introduction.- 1.1 The Problem.- 1.2 Prototype Problems.- 1.3 About this Book.- 1.4 Notes.- 2 Background.- 2.1 Real Analysis.- 2.2 Linear Algebra and Matrix Analysis.- 2.3 Numerical Linear Algebra.- 2.4 Convexity and Separation Theorems.- 2.5 Linear Equations and Inequalities.- 2.6 Convex Polyhedral Sets.- 2.7 Nonlinear System of Equations.- 2.8 Notes.- 3 Duality Theory and Optimality Conditions.- 3.1 The Dual Problem.- 3.2 Duality Theorems.- 3.3 Optimality and Complementary Slackness.- 3.4 Complementary Pair of Variables.- 3.5 Degeneracy and Uniqueness.- 3.6 Notes.- 4 Boundary Methods.- 4.1 Introduction.- 4.2 Primal Simplex Method.- 4.3 Bounded Variable Simplex Method.- 4.4 Dual Simplex Method.- 4.5 Primal — Dual Method.- 4.6 Notes.- 5 Interior Point Methods.- 5.1 Primal Affine Scaling Method.- 5.2 Degeneracy Resolution by Step-Size Control.- 5.3 Accelerated Affine Scaling Method.- 5.4 Primal Power Affine Scaling Method.- 5.5 Obtaining an Initial Interior Point.- 5.6 Bounded Variable Affine Scaling Method.- 5.7 Affine Scaling and Unrestricted Variables.- 5.8 Dual Affine Scaling Method.- 5.9 Primal-Dual Affine Scaling Method.- 5.10 Path Following or Homotopy Methods.- 5.11 Projective Transformation Method.- 5.12 Method and Unrestricted Variables.- 5.13 Notes.- 6 Implementation.- 6.1 Implementation of Boundary Methods.- 6.2 Implementation of Interior Point Methods.- 6.3 Notes.- A Tables.