Linear and Nonlinear Programming

By: David G. Luenberger, Yinyu Ye

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Hardcover | 7 July 2008 | Edition Number 3

At a Glance

Hardcover
564 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 3.18

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"Linear and Nonlinear Programming" is considered a classic textbook in Optimization. While it is a classic, it also reflects modern theoretical insights. These insights provide structure to what might otherwise be simply a collection of techniques and results, and this is valuable both as a means for learning existing material and for developing new results. One major insight of this type is the connection between the purely analytical character of an optimization problem, expressed perhaps by properties of the necessary conditions, and the behavior of algorithms used to solve a problem. This was a major theme of the first and second editions. Now the third edition has been completely updated with recent Optimization Methods. Yinyu Ye has written chapters and chapter material on a number of these areas including Interior Point Methods.

This book is designed for either self-study by professionals or classroom work at the undergraduate or graduate level for technical students. Like the field of optimization itself, which involves many classical disciplines, the book should be useful to system analysts, operations researchers, numerical analysts, management scientists, and other specialists.

Industry Reviews

From the reviews of the third edition:

"The aim of this book is to present the central concepts of optimization techniques. ... the material in this new edition is organized into three separate parts. ... this very well-written book is a classic textbook in Optimization. It should be present in the bookcase of each student, researcher, and specialist from the host of disciplines from which practical optimization applications are drawn." (Jean-Jacques Strodiot, Zentralblatt MATH, Vol. 1207, 2011)

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Introduction	p. 1
Optimization	p. 1
Types of Problems	p. 2
Size of Problems	p. 5
Iterative Algorithms and Convergence	p. 6
Linear Programming
Basic Properties of Linear Programs	p. 11
Introduction	p. 11
Examples of Linear Programming Problems	p. 14
Basic Solutions	p. 19
The Fundamental Theorem of Linear Programming	p. 20
Relations to Convexity	p. 22
Exercises	p. 28
The Simplex Method	p. 33
Pivots	p. 33
Adjacent Extreme Points	p. 38
Determining a Minimum Feasible Solution	p. 42
Computational Procedure-Simplex Method	p. 46
Artificial Variables	p. 50
Matrix Form of the Simplex Method	p. 54
The Revised Simplex Method	p. 56
The Simplex Method and LU Decomposition	p. 59
Decomposition	p. 62
Summary	p. 70
Exercises	p. 70
Duality	p. 79
Dual Linear Programs	p. 79
The Duality Theorem	p. 82
Relations to the Simplex Procedure	p. 84
Sensitivity and Complementary Slackness	p. 88
The Dual Simplex Method	p. 90
The Primal-Dual Algorithm	p. 93
Reduction of Linear Inequalities	p. 98
Exercises	p. 103
Interior-Point Methods	p. 111
Elements of Complexity Theory	p. 112
The Simplex Method is not Polynomial-Time	p. 114
The Ellipsoid Method	p. 115
The Analytic Center	p. 118
The Central Path	p. 121
Solution Strategies	p. 126
Termination and Initialization	p. 134
Summary	p. 139
Exercises	p. 140
Transportation and Network Flow Problems	p. 145
The Transportation Problem	p. 145
Finding a Basic Feasible Solution	p. 148
Basis Triangularity	p. 150
Simplex Method for Transportation Problems	p. 153
The Assignment Problem	p. 159
Basic Network Concepts	p. 160
Minimum Cost Flow	p. 162
Maximal Flow	p. 166
Summary	p. 174
Exercises	p. 175
Unconstrained Problems
Basic Properties of Solutions and Algorithms	p. 183
First-Order Necessary Conditions	p. 184
Examples of Unconstrained Problems	p. 186
Second-Order Conditions	p. 190
Convex and Concave Functions	p. 192
Minimization and Maximization of Convex Functions	p. 197
Zero-Order Conditions	p. 198
Global Convergence of Descent Algorithms	p. 201
Speed of Convergence	p. 208
Summary	p. 212
Exercises	p. 213
Basic Descent Methods	p. 215
Fibonacci and Golden Section Search	p. 216
Line Search by Curve Fitting	p. 219
Global Convergence of Curve Fitting	p. 226
Closedness of Line Search Algorithms	p. 228
Inaccurate Line Search	p. 230
The Method of Steepest Descent	p. 233
Applications of the Theory	p. 242
Newton's Method	p. 246
Coordinate Descent Methods	p. 253
Spacer Steps	p. 255
Summary	p. 256
Exercises	p. 257
Conjugate Direction Methods	p. 263
Conjugate Directions	p. 263
Descent Properties of the Conjugate Direction Method	p. 266
The Conjugate Gradient Method	p. 268
The C-G Method as an Optimal Process	p. 271
The Partial Conjugate Gradient Method	p. 273
Extension to Nonquadratic Problems	p. 277
Parallel Tangents	p. 279
Exercises	p. 282
Quasi-Newton Methods	p. 285
Modified Newton Method	p. 285
Construction of the Inverse	p. 288
Davidon-Fletcher-Powell Method	p. 290
The Broyden Family	p. 293
Convergence Properties	p. 296
Scaling	p. 299
Memoryless Quasi-Newton Methods	p. 304
Combination of Steepest Descent and Newton's Method	p. 306
Summary	p. 312
Exercises	p. 313
Constrained Minimization
Constrained Minimization Conditions	p. 321
Constraints	p. 321
Tangent Plane	p. 323
First-Order Necessary Conditions (Equality Constraints)	p. 326
Examples	p. 327
Second-Order Conditions	p. 333
Eigenvalues in Tangent Subspace	p. 335
Sensitivity	p. 339
Inequality Constraints	p. 341
Zero-Order Conditions and Lagrange Multipliers	p. 346
Summary	p. 353
Exercises	p. 354
Primal Methods	p. 359
Advantage of Primal Methods	p. 359
Feasible Direction Methods	p. 360
Active Set Methods	p. 363
The Gradient Projection Method	p. 367
Convergence Rate of the Gradient Projection Method	p. 374
The Reduced Gradient Method	p. 382
Convergence Rate of the Reduced Gradient Method	p. 387
Variations	p. 394
Summary	p. 396
Exercises	p. 396
Penalty and Barrier Methods	p. 401
Penalty Methods	p. 402
Barrier Methods	p. 405
Properties of Penalty and Barrier Functions	p. 407
Newton's Method and Penalty Functions	p. 416
Conjugate Gradients and Penalty Methods	p. 418
Normalization of Penalty Functions	p. 420
Penalty Functions and Gradient Projection	p. 421
Exact Penalty Functions	p. 425
Summary	p. 429
Exercises	p. 430
Dual and Cutting Plane Methods	p. 435
Global Duality	p. 435
Local Duality	p. 441
Dual Canonical Convergence Rate	p. 446
Separable Problems	p. 447
Augmented Lagrangians	p. 451
The Dual Viewpoint	p. 456
Cutting Plane Methods	p. 460
Kelley's Convex Cutting Plane Algorithm	p. 463
Modifications	p. 465
Exercises	p. 466
Primal-Dual Methods	p. 469
The Standard Problem	p. 469
Strategies	p. 471
A Simple Merit Function	p. 472
Basic Primal-Dual Methods	p. 474
Modified Newton Methods	p. 479
Descent Properties	p. 481
Rate of Convergence	p. 485
Interior Point Methods	p. 487
Semidefinite Programming	p. 491
Summary	p. 498
Exercises	p. 499
Mathematical Review	p. 507
Sets	p. 507
Matrix Notation	p. 508
Spaces	p. 509
Eigenvalues and Quadratic Forms	p. 510
Topological Concepts	p. 511
Functions	p. 512
Convex Sets	p. 515
Basic Definitions	p. 515
Hyperplanes and Polytopes	p. 517
Separating and Supporting Hyperplanes	p. 519
Extreme Points	p. 521
Gaussian Elimination	p. 523
Bibliography	p. 527
Index	p. 541
Table of Contents provided by Ingram. All Rights Reserved.

Linear and Nonlinear Programming

At a Glance

Hardcover

Industry Reviews

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How to return your order

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