Synopses & Reviews
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. Audience This book is addressed to mathematicians, engineers, economists, and researchers interested in acquiring a solid mathematical foundation in topics such as point-to-set operators, variational inequalities, general equilibrium theory, and nonsmooth optimization, among others. Containing extensive exercises and examples throughout the text, the first four chapters of the book can also be used for a one-quarter course in set-valued analysis and maximal monotone operators for graduate students in pure and applied mathematics, mathematical economics, operations research and related areas. The only requisites, besides a minimum level of mathematical maturity, are some basic results of general topology and functional analysis.
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)
This book gives a clear presentation of set-valued analysis and monotone operators and also presents several important new results in the field. The first part of the book begins with a self-contained and encompassing overview of several key topics within set-valued analysis. The authors present basic notions of set convergence, and of continuity of set-valued mappings, together with the 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.
The second part of the monograph contains new results, all of them established during the last decade, on a concept of enlargements of monotone operators, which is then applied to several fields of interest in applied mathematcs like bundle-type methods, augmented Lagrangian methods or variational inequalities and recently developed versions of proximal point algorithms. The results in the second part use extensively the concepts presented in the first one.
The unique features of the book include the previously unpublished contents of chapters 5-6, as well as the presentation of the results in chapters 1-4, which are at the same time accessible, self contained, and hold at a high level of generality.
All the concepts and practically all proofs are given within the text, so that a wide audience can have a friendly, but at the same time deep, access to many of the most relevant topics set-valued analysis.
This is the first comprehensive book treatment of the emerging subdiscipline of set-valued mapping and enlargements of maximal monotone operators. It features several important new results and applications in the field. Throughout the text, examples help readers make the bridge from theory to application. Numerous exercises are also offered to enable readers to apply and build their own skills and knowledge.
Table of Contents
List of Figures.- Dedication.- Acknowledgments.- Preface.- Introduction.- Set Convergence and Point-to-Set Mappings.- Convex Analysis and Fixed Point Theorems.- Maximal Monotone Operators.- 5. Enlargements of Monotone Operators.- Recent Topics in Proximal Theory.- Bibliography.- Notation.- Index.