Nonlinear Optimization

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Overview

"Nonlinear Optimization will become the standard textbook on its subject, as well as a reference book that everyone will want to own. Not only is it beautiful and elegant, it is also utterly comprehensive and modern, with many realistic and interesting examples."--Robert J. Vanderbei, Princeton University, author of Linear Programming

"This excellent book is the best I have reviewed in the past ten years. Very well written, its three main strengths are its treatment of theory and algorithms on equal terms, its mathematically driven presentation of the material, and its interesting examples and applications."--Ekkehard W. Sachs, Virginia Tech and Universitt Trier

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Editorial Reviews

Zentralblatt MATH Database Database
This is one of the best textbooks on nonlinear optimization I know. Focus is on both theory and algorithmic solution of convex as well as of differentiable programming problems.
— Stephan Dempe
Zentralblatt MATH Database
This is one of the best textbooks on nonlinear optimization I know. Focus is on both theory and algorithmic solution of convex as well as of differentiable programming problems.
— Stephan Dempe
Zentralblatt MATH Database - Stephan Dempe
This is one of the best textbooks on nonlinear optimization I know. Focus is on both theory and algorithmic solution of convex as well as of differentiable programming problems.
Operations Research Letters - Franz Rendl
In summary, this book competes with the topmost league of books on optimization. The wide range of topics covered and the thorough theoretical treatment of algorithms make it not only a good prospective textbook, but even more a reference text (which I am happy to have on my shelf.)
Mathematical Methods of Operation Research - Petri Eskelinen
Throughout the book the writing style is very clear, compact and easy to follow, but at the same time mathematically rigorous. The proofs are easy to follow because the author usually carefully explains every move. In addition the meaning of the most central results is usually demonstrated with examples and in many cases explanations are also supported by visualizations...This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods...Recommended as a material for both self study and teaching purposes
From the Publisher

"This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods. With no doubt the major strength of this book is the clear and intuitive structure and systematic style of presentation. This book can be recommended as a material for both self study and teaching purposes, but because of its rigorous style it works also as a valuable reference for research purposes."--Mathematical Modeling and Operational Research

"This is one of the best textbooks on nonlinear optimization I know. Focus is on both theory and algorithmic solution of convex as well as of differentiable programming problems."--Stephan Dempe, Zentralblatt MATH Database

"In summary, this book competes with the topmost league of books on optimization. The wide range of topics covered and the thorough theoretical treatment of algorithms make it not only a good prospective textbook, but even more a reference text (which I am happy to have on my shelf.)"--Franz Rendl, Operations Research Letters

"Throughout the book the writing style is very clear, compact and easy to follow, but at the same time mathematically rigorous. The proofs are easy to follow because the author usually carefully explains every move. In addition the meaning of the most central results is usually demonstrated with examples and in many cases explanations are also supported by visualizations...This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods...Recommended as a material for both self study and teaching purposes"--Petri Eskelinen, Mathematical Methods of Operation Research

Mathematical Modeling and Operational Research
This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods. With no doubt the major strength of this book is the clear and intuitive structure and systematic style of presentation. This book can be recommended as a material for both self study and teaching purposes, but because of its rigorous style it works also as a valuable reference for research purposes.
Operations Research Letters
In summary, this book competes with the topmost league of books on optimization. The wide range of topics covered and the thorough theoretical treatment of algorithms make it not only a good prospective textbook, but even more a reference text (which I am happy to have on my shelf.)
— Franz Rendl
Mathematical Methods of Operation Research
Throughout the book the writing style is very clear, compact and easy to follow, but at the same time mathematically rigorous. The proofs are easy to follow because the author usually carefully explains every move. In addition the meaning of the most central results is usually demonstrated with examples and in many cases explanations are also supported by visualizations...This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods...Recommended as a material for both self study and teaching purposes
— Petri Eskelinen
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Product Details

  • ISBN-13: 9780691119151
  • Publisher: Princeton University Press
  • Publication date: 1/2/2006
  • Edition description: New Edition
  • Pages: 464
  • Product dimensions: 9.30 (w) x 6.60 (h) x 1.40 (d)

Meet the Author

Andrzej Ruszczynski is Professor of Operations Research at Rutgers University. He is the coauthor of "Stochastic Programming" and the coeditor of "Decision Making under Uncertainty".

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Table of Contents

Preface xi

Chapter 1. Introduction 1

PART 1. THEORY 15

Chapter 2. Elements of Convex Analysis 17
2.1 Convex Sets 17
2.2 Cones 25
2.3 Extreme Points 39
2.4 Convex Functions 44
2.5 Subdifferential Calculus 57
2.6 Conjugate Duality 75

Chapter 3. Optimality Conditions 88
3.1 Unconstrained Minima of Differentiable Functions 88
3.2 Unconstrained Minima of Convex Functions 92
3.3 Tangent Cones 98
3.4 Optimality Conditions for Smooth Problems 113
3.5 Optimality Conditions for Convex Problems 125
3.6 Optimality Conditions for Smooth-Convex Problems 133
3.7 Second Order Optimality Conditions 139
3.8 Sensitivity 150

Chapter 4. Lagrangian Duality 160
4.1 The Dual Problem 160
4.2 Duality Relations 166
4.3 Conic Programming 175
4.4 Decomposition 180
4.5 Convex Relaxation of Nonconvex Problems 186
4.6 The Optimal Value Function 191
4.7 The Augmented Lagrangian 196

PART 2. METHODS 209

Chapter 5. Unconstrained Optimization of Differentiable Functions 211
5.1 Introduction to Iterative Algorithms 211
5.2 Line Search 213
5.3 The Method of Steepest Descent 218
5.4 Newton's Method 233
5.5 The Conjugate Gradient Method 240
5.6 Quasi-Newton Methods 257
5.7 Trust Region Methods 266
5.8 Nongradient Methods 275

Chapter 6. Constrained Optimization of Differentiable Functions 286
6.1 Feasible Point Methods 286
6.2 Penalty Methods 297
6.3 The Basic Dual Method 308
6.4 The Augmented Lagrangian Method 311
6.5 Newton's Method 324
6.6 Barrier Methods 331

Chapter 7. Nondifferentiable Optimization 343
7.1 The Subgradient Method 343
7.2 The Cutting Plane Method 357
7.3 The Proximal Point Method 366
7.4 The Bundle Method 372
7.5 The Trust Region Method 384
7.6 Constrained Problems 389
7.7 Composite Optimization 397
7.8 Nonconvex Constraints 406

Appendix A. Stability of Set-Constrained Systems 411
A.1 Linear-Conic Systems 411
A.2 Set-Constrained Linear Systems 415
A.3 Set-Constrained Nonlinear Systems 418
Further Reading 427

Bibliography 431
Index 445

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Sort by: Showing 1 Customer Reviews
  • Anonymous

    Posted November 10, 2006

    Excellent introduction to theory and algorithms for nonlinear optimization

    This outstanding book fills the need for a recent introductory graduate textbook in nonlinear convex optimization. The book is divided into 2 parts: Part I deals with theory while Part II deals with algorithms for nonlinear convex optimization. Topics covered in Part I include basic convex analysis, optimality conditions, and Lagrangian duality. There are a number of interesting examples distributed throughout the discussions in Part I - some of these examples include recent concepts like semidefinite programming. The author also highlights the importance of DIFFERENTIABILITY in convex optimization - in fact he devotes separate sections for the optimality conditions of smooth convex and nonsmooth convex problems. Part II discusses algorithms for smooth unconstrained and constrained optimization and finally subgradient, bundle, and trust region schemes for nondifferentiable optimization. The discussion on algorithms for nondifferentiable optimization is new and an important ingredient in this book - for more details one can refer to the 2 volume set by Hiriart-Urruty and Lemarechal. However, there is no discussion on INTERIOR POINT METHODS and this is the only notable omission in the book. For more on interior point methods in nonlinear optimization, one can refer to the recent book by Nocedal and Wright. Personally, I enjoyed this book immensely, and I look forward to using it in a graduate course on nonlinear optimization.

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