Optimization is one of the most important areas of modern applied mathematics, with applications in fields from engineering and economics to finance, statistics, management science, and medicine. While many books have addressed its various aspects, "Nonlinear Optimization" is the first comprehensive treatment that will allow graduate students and researchers to understand its modern ideas, principles, and methods within a reasonable time, but without sacrificing mathematical precision. Andrzej Ruszczynski, a leading expert in the optimization of nonlinear stochastic systems, integrates the theory and the methods of nonlinear optimization in a unified, clear, and mathematically rigorous fashion, with detailed and easy-to-follow proofs illustrated by numerous examples and figures.
The book covers convex analysis, the theory of optimality conditions, duality theory, and numerical methods for solving unconstrained and constrained optimization problems. It addresses not only classical material but also modern topics such as optimality conditions and numerical methods for problems involving nondifferentiable functions, semidefinite programming, metric regularity and stability theory of set-constrained systems, and sensitivity analysis of optimization problems.
Based on a decade's worth of notes the author compiled in successfully teaching the subject, this book will help readers to understand the mathematical foundations of the modern theory and methods of nonlinear optimization and to analyze new problems, develop optimality theory for them, and choose or construct numerical solution methods. It is a must for anyone seriously interested in optimization.
"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
Preface xiChapter 1. Introduction 1PART 1. THEORY 15Chapter 2. Elements of Convex Analysis 172.1 Convex Sets 172.2 Cones 252.3 Extreme Points 392.4 Convex Functions 442.5 Subdifferential Calculus 572.6 Conjugate Duality 75Chapter 3. Optimality Conditions 883.1 Unconstrained Minima of Differentiable Functions 883.2 Unconstrained Minima of Convex Functions 923.3 Tangent Cones 983.4 Optimality Conditions for Smooth Problems 1133.5 Optimality Conditions for Convex Problems 1253.6 Optimality Conditions for Smooth-Convex Problems 1333.7 Second Order Optimality Conditions 1393.8 Sensitivity 150Chapter 4. Lagrangian Duality 1604.1 The Dual Problem 1604.2 Duality Relations 1664.3 Conic Programming 1754.4 Decomposition 1804.5 Convex Relaxation of Nonconvex Problems 1864.6 The Optimal Value Function 1914.7 The Augmented Lagrangian 196PART 2. METHODS 209Chapter 5. Unconstrained Optimization of Differentiable Functions 2115.1 Introduction to Iterative Algorithms 2115.2 Line Search 2135.3 The Method of Steepest Descent 2185.4 Newton's Method 2335.5 The Conjugate Gradient Method 2405.6 Quasi-Newton Methods 2575.7 Trust Region Methods 2665.8 Nongradient Methods 275Chapter 6. Constrained Optimization of Differentiable Functions 2866.1 Feasible Point Methods 2866.2 Penalty Methods 2976.3 The Basic Dual Method 3086.4 The Augmented Lagrangian Method 3116.5 Newton's Method 3246.6 Barrier Methods 331Chapter 7. Nondifferentiable Optimization 3437.1 The Subgradient Method 3437.2 The Cutting Plane Method 3577.3 The Proximal Point Method 3667.4 The Bundle Method 3727.5 The Trust Region Method 3847.6 Constrained Problems 3897.7 Composite Optimization 3977.8 Nonconvex Constraints 406Appendix A. Stability of Set-Constrained Systems 411A.1 Linear-Conic Systems 411A.2 Set-Constrained Linear Systems 415A.3 Set-Constrained Nonlinear Systems 418Further Reading 427Bibliography 431Index 445
Series: International Studen
Tertiary; University or College
Number Of Pages: 464
Published: 22nd January 2006
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2
Weight (kg): 0.79