Set-Valued Mappings and Enlargements of Monotone Operators: (8 Springer Optimization and Its Applications)

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.

Table of Contents:
Set Convergence and Point-to-Set Mappings.- Convex Analysis and Fixed Point Theorems.- Maximal Monotone Operators.- Enlargements of Monotone Operators.- Recent Topics in Proximal Theory.

Review :
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)