Convex Analysis and Nonlinear Optimization

Theory and Examples

By: Jonathan Borwein, Adrian S. Lewis

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Hardcover | 7 February 2006 | Edition Number 2

At a Glance

Hardcover
328 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 1.91

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Optimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained.

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MATHEMATICAL REVIEWS

"The present book gives a concise treatment of the area, aiming to show the relevance in particular of new developments in nonsmooth analysis to optimization theory...The book is of a manageable size and as such should appeal to the student. Further, the proofs are generally short and snappy, revealing the power of the abstract structural approach and fruitful interplay of geometrical and topological ideas. However, considerable ground is covered and, as a graduate text should, it develops the subject up to the frontiers of current research, giving an idea of areas for further exploration...This text will give impetus to the teaching of analysis because it makes evident its significant applications in optimization. But it will also bring added attraction to the study of optimization because it reveals so much of its abstract structural base."

"The book is divided into 11 chapters and provides a comprehensive presentation of the main features of convex analysis and nonlinear optimisation. Each result is sustained by a set of theorems, propositions and corollaries and includes rigorous proofs and clarifying discussions. They are complemented by a series of theoretical exercises. ... This book is warmly recommended for an advanced course in analysis for mathematicians or as a first graduate course for students involved with optimization theory." (Carlos Narciso Bouza Herrera, Zentralblatt MATH, Vol. 1116 (18), 2007)

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Preface	p. vii
Background	p. 1
Euclidean Spaces	p. 1
Symmetric Matrices	p. 9
Inequality Constraints	p. 15
Optimality Conditions	p. 15
Theorems of the Alternative	p. 23
Max-functions	p. 28
Fenchel Duality	p. 33
Subgradients and Convex Functions	p. 33
The Value Function	p. 43
The Fenchel Conjugate	p. 49
Convex Analysis	p. 65
Continuity of Convex Functions	p. 65
Fenchel Biconjugation	p. 76
Lagrangian Duality	p. 88
Special Cases	p. 97
Polyhedral Convex Sets and Functions	p. 97
Functions of Eigenvalues	p. 104
Duality for Linear and Semidefinite Programming	p. 109
Convex Process Duality	p. 114
Nonsmooth Optimization	p. 123
Generalized Derivatives	p. 123
Regularity and Strict Differentiability	p. 130
Tangent Cones	p. 137
The Limiting Subdifferential	p. 145
Karush-Kuhn-Tucker Theory	p. 153
An Introduction to Metric Regularity	p. 153
The Karush-Kuhn-Tucker Theorem	p. 160
Metric Regularity and the Limiting Subdifferential	p. 166
Second Order Conditions	p. 172
Fixed Points	p. 179
The Brouwer Fixed Point Theorem	p. 179
Selection and the Kakutani-Fan Fixed Point Theorem	p. 190
Variational Inequalities	p. 200
More Nonsmooth Structure	p. 213
Rademacher's Theorem	p. 213
Proximal Normals and Chebyshev Sets	p. 218
Amenable Sets and Prox-Regularity	p. 228
Partly Smooth Sets	p. 233
Postscript: Infinite Versus Finite Dimensions	p. 239
Introduction	p. 239
Finite Dimensionality	p. 241
Counterexamples and Exercises	p. 244
Notes on Previous Chapters	p. 248
List of Results and Notation	p. 253
Named Results	p. 253
Notation	p. 267
Bibliography	p. 275
Index	p. 289
Table of Contents provided by Ingram. All Rights Reserved.

Convex Analysis and Nonlinear Optimization

Theory and Examples

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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