Linear Programming

Foundations and Extensions

By: Robert J. Vanderbei

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Hardcover | 31 December 1996

At a Glance

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440 Pages

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24.13 x 16.51 x 2.54

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This book focuses largely on constrained optimization. It begins with a substantial treatment of linear programming and proceeds to convex analysis, network flows, integer programming, quadratic programming, and convex optimization. Along the way, dynamic programming and the linear complementarity problem are touched on as well.
This book aims to be the first introduction to the topic. Specific examples and concrete algorithms precede more abstract topics. Nevertheless, topics covered are developed in some depth, a large number of numerical examples worked out in detail, and many recent results are included, most notably interior-point methods. The exercises at the end of each chapter both illustrate the theory, and, in some cases, extend it.
Optimization is not merely an intellectual exercise: its purpose is to solve practical problems on a computer. Accordingly, the book comes with software that implements the major algorithms studied. At this point, software for the following four algorithms is available:

The two-phase simplex method

The primal-dual simplex method

The path-following interior-point method

The homogeneous self-dual methods.£/LIST£.

Industry Reviews

`Vanderbei's book is thoroughly modern. Vanderbei's book has many novel features. Some nice features. This book has style. Overall, I greatly enjoyed reviewing this book, and I highly recommend it as a textbook for an advanced undergraduate or master's level course in linear programming, particularly for courses in an engineering environment. In addition, it also is a good reference book for interior point methods as well as for implementation and computational aspects of linear programming. This is an excellent new book.'
Robert Freund, (MIT) in Optima, 56 (1997)
`In conclusion, Vanderbei's book gives an excellent introduction to linear programminbg, especially the algorithmic side of the subject. The book is highly recommended for both self study and as teaching material.'
Optima, 58 (1998)

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You Can Find This Book In

Non-Fiction Mathematics Optimisation Linear Programming Business & Management Operational Research Engineering & Technology Mechanical Engineering & Materials Production & Industrial Engineering Applied Mathematics

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Paperback

Published: 31st March 1998

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Preface	p. xiii
Preface to 2nd Edition	p. xvii
Basic Theory--The Simplex Method and Duality	p. 1
Introduction	p. 3
Managing a Production Facility	p. 3
The Linear Programming Problem	p. 6
Exercises	p. 8
Notes	p. 10
The Simplex Method	p. 13
An Example	p. 13
The Simplex Method	p. 16
Initialization	p. 19
Unboundedness	p. 22
Geometry	p. 22
Exercises	p. 24
Notes	p. 27
Degeneracy	p. 29
Definition of Degeneracy	p. 29
Two Examples of Degenerate Problems	p. 29
The Perturbation/Lexicographic Method	p. 32
Bland's Rule	p. 36
Fundamental Theorem of Linear Programming	p. 38
Geometry	p. 39
Exercises	p. 42
Notes	p. 43
Efficiency of the Simplex Method	p. 45
Performance Measures	p. 45
Measuring the Size of a Problem	p. 45
Measuring the Effort to Solve a Problem	p. 46
Worst-Case Analysis of the Simplex Method	p. 47
Exercises	p. 52
Notes	p. 53
Duality Theory	p. 55
Motivation--Finding Upper Bounds	p. 55
The Dual Problem	p. 57
The Weak Duality Theorem	p. 58
The Strong Duality Theorem	p. 60
Complementary Slackness	p. 66
The Dual Simplex Method	p. 68
A Dual-Based Phase I Algorithm	p. 71
The Dual of a Problem in General Form	p. 73
Resource Allocation Problems	p. 74
Lagrangian Duality	p. 78
Exercises	p. 79
Notes	p. 87
The Simplex Method in Matrix Notation	p. 89
Matrix Notation	p. 89
The Primal Simplex Method	p. 91
An Example	p. 96
The Dual Simplex Method	p. 101
Two-Phase Methods	p. 104
Negative Transpose Property	p. 105
Exercises	p. 108
Notes	p. 109
Sensitivity and Parametric Analyses	p. 111
Sensitivity Analysis	p. 111
Parametric Analysis and the Homotopy Method	p. 115
The Parametric Self-Dual Simplex Method	p. 119
Exercises	p. 120
Notes	p. 124
Implementation Issues	p. 125
Solving Systems of Equations: LU-Factorization	p. 126
Exploiting Sparsity	p. 130
Reusing a Factorization	p. 136
Performance Tradeoffs	p. 140
Updating a Factorization	p. 141
Shrinking the Bump	p. 145
Partial Pricing	p. 146
Steepest Edge	p. 147
Exercises	p. 149
Notes	p. 150
Problems in General Form	p. 151
The Primal Simplex Method	p. 151
The Dual Simplex Method	p. 153
Exercises	p. 159
Notes	p. 160
Convex Analysis	p. 161
Convex Sets	p. 161
Caratheodory's Theorem	p. 163
The Separation Theorem	p. 165
Farkas' Lemma	p. 167
Strict Complementarity	p. 168
Exercises	p. 170
Notes	p. 171
Game Theory	p. 173
Matrix Games	p. 173
Optimal Strategies	p. 175
The Minimax Theorem	p. 177
Poker	p. 181
Exercises	p. 184
Notes	p. 187
Regression	p. 189
Measures of Mediocrity	p. 189
Multidimensional Measures: Regression Analysis	p. 191
L[superscript 2]-Regression	p. 193
L[superscript 1]-Regression	p. 195
Iteratively Reweighted Least Squares	p. 196
An Example: How Fast is the Simplex Method?	p. 198
Which Variant of the Simplex Method is Best?	p. 202
Exercises	p. 203
Notes	p. 208
Network-Type Problems	p. 211
Network Flow Problems	p. 213
Networks	p. 213
Spanning Trees and Bases	p. 216
The Primal Network Simplex Method	p. 221
The Dual Network Simplex Method	p. 225
Putting It All Together	p. 228
The Integrality Theorem	p. 231
Exercises	p. 232
Notes	p. 240
Applications	p. 241
The Transportation Problem	p. 241
The Assignment Problem	p. 243
The Shortest-Path Problem	p. 244
Upper-Bounded Network Flow Problems	p. 247
The Maximum-Flow Problem	p. 250
Exercises	p. 252
Notes	p. 257
Structural Optimization	p. 259
An Example	p. 259
Incidence Matrices	p. 261
Stability	p. 262
Conservation Laws	p. 264
Minimum-Weight Structural Design	p. 267
Anchors Away	p. 269
Exercises	p. 272
Notes	p. 272
Interior-Point Methods	p. 275
The Central Path	p. 277
Warning: Nonstandard Notation Ahead	p. 277
The Barrier Problem	p. 277
Lagrange Multipliers	p. 280
Lagrange Multipliers Applied to the Barrier Problem	p. 283
Second-Order Information	p. 285
Existence	p. 285
Exercises	p. 287
Notes	p. 289
A Path-Following Method	p. 291
Computing Step Directions	p. 291
Newton's Method	p. 293
Estimating an Appropriate Value for the Barrier Parameter	p. 294
Choosing the Step Length Parameter	p. 295
Convergence Analysis	p. 296
Exercises	p. 302
Notes	p. 306
The KKT System	p. 307
The Reduced KKT System	p. 307
The Normal Equations	p. 308
Step Direction Decomposition	p. 310
Exercises	p. 313
Notes	p. 313
Implementation Issues	p. 315
Factoring Positive Definite Matrices	p. 315
Quasidefinite Matrices	p. 319
Problems in General Form	p. 325
Exercises	p. 331
Notes	p. 331
The Affine-Scaling Method	p. 333
The Steepest Ascent Direction	p. 333
The Projected Gradient Direction	p. 335
The Projected Gradient Direction with Scaling	p. 337
Convergence	p. 341
Feasibility Direction	p. 343
Problems in Standard Form	p. 344
Exercises	p. 345
Notes	p. 346
The Homogeneous Self-Dual Method	p. 349
From Standard Form to Self-Dual Form	p. 349
Homogeneous Self-Dual Problems	p. 350
Back to Standard Form	p. 360
Simplex Method vs Interior-Point Methods	p. 363
Exercises	p. 367
Notes	p. 368
Extensions	p. 371
Integer Programming	p. 373
Scheduling Problems	p. 373
The Traveling Salesman Problem	p. 375
Fixed Costs	p. 378
Nonlinear Objective Functions	p. 378
Branch-and-Bound	p. 380
Exercises	p. 392
Notes	p. 393
Quadratic Programming	p. 395
The Markowitz Model	p. 395
The Dual	p. 399
Convexity and Complexity	p. 402
Solution Via Interior-Point Methods	p. 404
Practical Considerations	p. 406
Exercises	p. 409
Notes	p. 411
Convex Programming	p. 413
Differentiable Functions and Taylor Approximations	p. 413
Convex and Concave Functions	p. 414
Problem Formulation	p. 414
Solution Via Interior-Point Methods	p. 415
Successive Quadratic Approximations	p. 417
Merit Functions	p. 417
Parting Words	p. 421
Exercises	p. 421
Notes	p. 423
Source Listings	p. 425
The Self-Dual Simplex Method	p. 426
The Homogeneous Self-Dual Method	p. 429
Answers to Selected Exercises	p. 433
Bibliography	p. 435
Index	p. 443
Table of Contents provided by Syndetics. All Rights Reserved.

Linear Programming

Foundations and Extensions

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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