Techniques of Variational Analysis

By: Jonathan M. Borwein, Qiji Zhu

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Hardcover | 4 July 2005

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376 Pages

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Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse fields. This book provides a concise account of the essential tools of infinite-dimensional first-order variational analysis. These tools are illustrated by applications in many different parts of analysis, optimization and approximation, dynamical systems, mathematical economics and elsewhere. Much of the material in the book grows out of talks and short lecture series given by the authors in the past several years. Thus, chapters in this book can easily be arranged to form material for a graduate level topics course. A sizeable collection of suitable exercises is provided for this purpose. In addition, this book is also a useful reference for researchers who use variational techniques---or just think they might like to.

Industry Reviews

From the reviews:

"This book maps the progress that has been made since the publication of the Ekeland variational principle in 1974 in the development and application of the variational approach in nonlinear analysis. The authors are well equipped for their task. ... This monograph is distinctive for bringing out the unifying role of variational principles across nonlinear analysis, the numerous examples of their application, and for the insights communicated by the authors, drawing on their experience as key participants in their development." (Richard B. Vinter, Mathematical Reviews, Issue 2006 h)

"The book presents a nice treatment of known variational principles and their application in many fields of mathematics. ... Many exercises are provided at the end of all sections where the reader can reflect the main text and can get further generalizations of the results." (Joerg Thierfelder, Zentralblatt MATH, Vol. 1076, 2006)

"The aim of the book is to emphasize the strength of the variational techniques in various domains ... . The book contains a lot of exercises completing the main text ... . the book is directed to graduate students in the field of variational analysis. ... Researchers who use variational techniques or intend to do so, will find the book very useful too." (S. Cobzas, Studia Universitatis Babes-Bolyai Mathematica, Vol. LI (2), June, 2006)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Calculus of Variations Optimisation Linear Programming Applied Mathematics

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Published: 1st December 2010

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Introduction	p. 1
Introduction	p. 1
Notation	p. 2
Exercises	p. 4
Variational Principles	p. 5
Ekeland Variational Principles	p. 5
Geometric Forms of the Variational Principle	p. 10
Applications to Fixed Point Theorems	p. 15
Finite Dimensional Variational Principles	p. 19
Borwein-Preiss Variational Principles	p. 30
Variational Techniques in Subdifferential Theory	p. 37
The Frechet Subdifferential and Normal Cone	p. 39
Nonlocal Sum Rule and Viscosity Solutions	p. 47
Local Sum Rules and Constrained Minimization	p. 54
Mean Value Theorems and Applications	p. 78
Chain Rules and Lyapunov Functions	p. 87
Multidirectional MVI and Solvability	p. 95
Extremal Principles	p. 103
Variational Techniques in Convex Analysis	p. 111
Convex Functions and Sets	p. 111
Subdifferential	p. 117
Sandwich Theorems and Calculus	p. 127
Fenchel Conjugate	p. 134
Convex Feasibility Problems	p. 140
Duality Inequalities for Sandwiched Functions	p. 150
Entropy Maximization	p. 157
Variational Techniques and Multifunctions	p. 165
Multifunctions	p. 165
Subdifferentials as Multifunctions	p. 188
Distance Functions	p. 214
Coderivatives of Multifunctions	p. 220
Implicit Multifunction Theorems	p. 229
Variational Principles in Nonlinear Functional Analysis	p. 243
Subdifferential and Asplund Spaces	p. 243
Nonconvex Separation Theorems	p. 259
Stegall Variational Principles	p. 266
Mountain Pass Theorem	p. 274
One-Perturbation Variational Principles	p. 280
Variational Techniques in the Presence of Symmetry	p. 291
Nonsmooth Functions on Smooth Manifolds	p. 291
Manifolds of Matrices and Spectral Functions	p. 299
Convex Spectral Functions	p. 316
References	p. 339
Index	p. 353
Table of Contents provided by Ingram. All Rights Reserved.

Techniques of Variational Analysis

At a Glance

Hardcover

Industry Reviews

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You Can Find This Book In

Other Editions and Formats

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