Overview
- First comprehensive treatment in book form of the fairly new subdiscipline in set-valued analysis of enlargements of maximal monotone operators
- Presents several important new results in the field
- Addressed to graduate students, mathematicians, engineers, economists, and researchers
- Contains extensive exercises and examples throughout the text
Part of the book series: Springer Optimization and Its Applications (SOIA, volume 8)
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Table of contents (6 chapters)
Keywords
About this book
Set-valued analysis is an essential tool for the mathematical formulation of many real-life situations, e.g., equilibrium theory in mathematical economics. This work offers the first comprehensive treatment in book form of the fairly new subdiscipline of enlargements of maximal monotone operators, including several important new results in the field. In the last decades, with the development of nonsmooth optimization, effective algorithms have been developed to solve these kinds of problems, such as nonsmooth variational inequalities. Several of these methods, such as bundle methods for variational problems, are fully developed and analyzed in this book.
The first chapters provide a self-contained review of the basic notions and fundamental results in set-valued analysis, including set convergence and continuity of set-valued mappings together with many important results in infinite-dimensional convex analysis, leading to the classical fixed point results due to Ekeland, Caristi and Kakutani. Next, an in-depth introduction to monotone operators is developed, emphasizing results related to maximality of subdifferentials and of sums of monotone operators. Building on this foundational material, the second part of the monograph contains new results (all of them established during the last decade) on the concept of enlargements of monotone operators, with applications to variational inequalities, bundle-type methods, augmented Lagrangian methods, and proximal point algorithms.
Reviews
From the reviews:
"This book is a welcome new monograph on set-valued mappings. The emphasis is on enlargements of monotone operators. … The authors are major contributors to the research in this area. They give, with much insight, a systematic and efficient account of the state of the art of the recent research in this area … . Most of the sections are accompanied by exercises, making the book also a nice choice for a graduate topic course." (Qiji Jim Zhu, Mathematical Reviews, Issue 2008 h)
"The main aim of this book is to give a relatively brief and self-contained review of the basics of set-valued analysis in a Banach space, and to present most significant results related to existence of fixed points of multivalued mappings. … At the end of each chapter the authors give detailed historical notes and references for further reading. The volume can be read by anyone with a basic knowledge of real and functional analysis." (Mikhail Yu. Kokurin, Zentralblatt MATH, Vol. 1154, 2009)
Authors and Affiliations
Bibliographic Information
Book Title: Set-Valued Mappings and Enlargements of Monotone Operators
Authors: Regina S. Burachik, Alfredo N. Iusem
Series Title: Springer Optimization and Its Applications
DOI: https://doi.org/10.1007/978-0-387-69757-4
Publisher: Springer New York, NY
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag US 2008
Hardcover ISBN: 978-0-387-69755-0Published: 15 October 2007
Softcover ISBN: 978-1-4419-4346-0Published: 29 November 2010
eBook ISBN: 978-0-387-69757-4Published: 15 November 2007
Series ISSN: 1931-6828
Series E-ISSN: 1931-6836
Edition Number: 1
Number of Pages: XIV, 294
Number of Illustrations: 15 b/w illustrations
Topics: Functional Analysis, Optimization, Operator Theory, Operations Research, Management Science, Operations Research/Decision Theory, Calculus of Variations and Optimal Control; Optimization