Convex Analysis and Monotone Operator Theory in Hilbert Spaces

by Heinz H. Bauschke, Patrick L. Combettes

Overview

This book examines results of convex analysis and optimization in Hilbert space, presenting a concise exposition of related theory that allows for algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions and more.

Product Details

ISBN-13:	9783319483115
Publisher:	Springer-Verlag New York, LLC
Publication date:	02/28/2017
Series:	CMS Books in Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	619
File size:	63 MB
Note:	This product may take a few minutes to download.

About the Author

Heinz H. Bauschke is a Full Professor of Mathematics at the Kelowna campus of the University of British Columbia, Canada.

Patrick L. Combettes, IEEE Fellow, was on the faculty of the City University of New York and of Université Pierre et Marie Curie – Paris 6 before joining North Carolina State University as a Distinguished Professor of Mathematics in 2016.

Table of Contents

Background.- Hilbert Spaces.- Convex Sets.- Convexity and Notation of Nonexpansiveness.- Fejer Monotonicity and Fixed Point Iterations.- Convex Cones and Generalized Interiors.- Support Functions and Polar Sets.- Convex Functions.- Lower Semicontinuous Convex Functions.- Convex Functions: Variants.- Convex Minimization Problems.- Infimal Convolution.- Conjugation.- Further Conjugation Results.- Fenchel-Rockafellar Duality.- Subdifferentiability of Convex Functions.- Differentiability of Convex Functions.- Further Differentiability Results.- Duality in Convex Optimization.- Monotone Operators.- Finer Properties of Monotone Operators.- Stronger Notions of Monotonicity.- Resolvents of Monotone Operators.- Proximity Operators.- Sums of Monotone Operators.- Zeros of Sums of Monotone Operators.- Fermat's Rule in Convex Optimization.- Proximal Minimization.- Projection Operators.- Best Approximation Algorithms.

Editorial Reviews

From the reviews:

“This book is devoted to a review of basic results and applications of convex analysis, monotone operator theory, and the theory of nonexpansive mappings in Hilbert spaces. … Each chapter concludes with an exercise section. Bibliographical pointers, a summary of symbols and notation, an index, and a comprehensive reference list are also included. The book is suitable for graduate students and researchers in pure and applied mathematics, engineering and economics.” (Sergiu Aizicovici, Zentralblatt MATH, Vol. 1218, 2011)

“This timely, well-written, informative and readable book is a largely self-contained exposition of the main results … in Hilbert spaces. … The high level of the presentation, the careful and detailed discussion of many applications and algorithms, and last, but not least, the inclusion of more than four hundred exercises, make the book accessible and of great value to students, practitioners and researchers … .” (Simeon Reich, Mathematical Reviews, Issue 2012 h)

From the Publisher