Nonlinear Programming: Theory and Algorithms / Edition 2

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Overview

Presents recent developments of key topics in nonlinear programming using a logical and self-contained format. Divided into three sections that deal with convex analysis, optimality conditions and duality, computational techniques. Precise statements of algorithms are given along with convergence analysis. Each chapter contains detailed numerical examples, graphical illustrations and numerous exercises to aid readers in understanding the concepts and methods discussed.
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Editorial Reviews

Booknews
A self-contained text/reference dealing with convex analysis, optimality conditions and duality, and computational methods--presented with numerical examples, graphical illustrations, and exercises. Useful as a reference for topics in nonlinear programming and as a text in the fields of operations research, management science, industrial engineering, and applied math, and in engineering disciplines that deal with analytical optimization techniques. Requires some mathematical maturity and a working knowledge of linear algebra and calculus. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
"The promotional message on the back cover proclaims 'this book is a solid reference for professionals and a useful text for students…"; and I fully agree." (Technometrics, February 2007)

"Noted and recommended for its logical format and sharp editing that never wavers in its focus." (Electric Review, September/October 2006)

"…highly recommended for a course in the theory of nonlinear programming…" (MAA Reviews, July 17, 2006)

 ‘… ‘the Bazaraa’ is a must if you are interested in optimization…’ (Journal of the Operational Research Society, 2007)

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Product Details

  • ISBN-13: 9780471557937
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 11/1/1992
  • Series: Interscience Series in Discrete Mathematics
  • Edition description: Older Edition
  • Edition number: 2
  • Pages: 656
  • Product dimensions: 7.32 (w) x 10.23 (h) x 1.37 (d)

Meet the Author

Mokhtar S. BAZARAA, PhD, is a Professor at the Georgia Institute of Technology.

HANIF D. SHERALI, PhD, is a W. Thomas Rice Chaired Professor of Engineering in the Grado Department of Industrial and Systems Engineering at Virginia Polytechnic Institute and State University.

C. M. SHETTY, PhD, is a Professor Emeritus at the Georgia Institute of Technology.
Professors Bazaraa and Sherali are also coauthors of the complementary bestselling book, Linear Programming and Network Flows, Third Edition, also published by Wiley.

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Table of Contents

Ch. 1 Introduction 1
1.1 Problem Statement and Basic Definitions 2
1.2 Some Illustrative Examples 4
1.3 Some Guidelines for Model Construction 21
Pt. 1 Convex Analysis 31
Ch. 2 Convex Sets 33
2.1 Convex Hulls 34
2.2 Closure and Interior of a Set 38
2.3 Weierstrass' Theorem 41
2.4 Separation and Support of Sets 42
2.5 Convex Cones and Polarity 52
2.6 Polyhedral Sets, Extreme Points, and Extreme Directions 54
2.7 Linear Programming and the Simplex Method 62
Ch. 3 Convex Functions and Generalizations 78
3.1 Definitions and Basic Properties 79
3.2 Subgradients of Convex Functions 83
3.3 Differentiable Convex Functions 88
3.4 Minima and Maxima of Convex Functions 99
3.5 Generalizations of Convex Functions 107
Pt. 2 Optimality Conditions and Duality 129
Ch. 4 The Fritz John and the Karush-Kuhn-Tucker Optimality Conditions 131
4.1 Unconstrained Problems 132
4.2 Problems with Inequality Constraints 138
4.3 Problems with Inequality and Equality Constraints 156
4.4 Second-Order Necessary and Sufficient Optimality Conditions for Constrained Problems 167
Ch. 5 Constraint Qualifications 184
5.1 The Cone of Tangents 184
5.2 Other Constraint Qualifications 188
5.3 Problems with Inequality and Equality Constraints 191
Ch. 6 Lagrangian Duality and Saddle Point Optimality Conditions 199
6.1 The Lagrangian Dual Problem 200
6.2 Duality Theorems and Saddle Point Optimality Conditions 203
6.3 Properties of the Dual Function 214
6.4 Formulating and Solving the Dual Problem 222
6.5 Getting the Primal Solution 228
6.6 Linear and Quadratic Programs 231
Pt. 3 Algorithms and Their Convergence 243
Ch. 7 The Concept of an Algorithm 244
7.1 Algorithms and Algorithmic Maps 244
7.2 Closed Maps and Convergence 247
7.3 Composition of Mappings 251
7.4 Comparison Among Algorithms 255
Ch. 8 Unconstrained Optimization 265
8.1 Line Search Without Using Derivatives 266
8.2 Line Search Using Derivatives 276
8.3 Some Practical Line Search Methods 279
8.4 Closedness of the Line Search Algorithmic Map 282
8.5 Multidimensional Search Without Using Derivatives 283
8.6 Multidimensional Search Using Derivatives 300
8.7 Modification of Newton's Method: Levenberg-Marquardt and Trust Region Methods 312
8.8 Methods Using Conjugate Directions: Quasi-Newton and Conjugate Gradient Methods 315
8.9 Subgradient Optimization Methods 339
Ch. 9 Penalty and Barrier Functions 360
9.1 The Concept of Penalty Functions 361
9.2 Exterior Penalty Function Methods 365
9.3 Exact Absolute Value and Augmented Lagrangian Penalty Methods 372
9.4 Barrier Function Methods 385
9.5 A Polynomial-Time Algorithm for Linear Programming Based on a Barrier Function 392
Ch. 10 Methods of Feasible Directions 408
10.1 The Method of Zoutendijk 409
10.2 Convergence Analysis of the Method of Zoutendijk 423
10.3 Successive Linear Programming Approach 432
10.4 Successive Quadratic Programming or Projected Lagrangian Approach 438
10.5 The Gradient Projection Method of Rosen 448
10.6 The Method of Reduced Gradient of Wolfe and the Generalized Reduced Gradient Method 458
10.7 The Convex-Simplex Method of Zangwill 466
10.8 Effective First- and Second-Order Variants of the Reduced Gradient Method 471
Ch. 11 Linear Complementary Problem, and Quadratic, Separable, Fractional, and Geometric Programming 493
11.1 The Linear Complementary Problem 494
11.2 Quadratic Programming 503
11.3 Separable Programming 509
11.4 Linear Fractional Programming 524
11.5 Geometric Programming 531
Appendix A. Mathematical Review 554
Appendix B. Summary of Convexity, Optimality Conditions, and Duality 565
Bibliography 576
Index 627
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