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Springer Optimization and Its Applications #8: Set-Valued Mappings and Enlargements of Monotone Operatorsby Regina Burachik
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.
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.
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.
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.
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