Convex Analysis and Beyond

Volume I: Basic Theory
 
Edition number: 1st ed. 2022
Publisher: Springer
Date of Publication:
Number of Volumes: 1 pieces, Book
 
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Product details:

ISBN13:9783030947873
ISBN10:3030947874
Binding:Paperback
No. of pages:585 pages
Size:235x155 mm
Weight:1070 g
Language:English
Illustrations: 7 Illustrations, black & white; 40 Illustrations, color
606
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Short description:

This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications.
 
The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.   

Long description:
This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications.
 
The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.


?Each chapter ends with an exercise section ? . While primarily addressed to researchers, the book can be used for graduate courses in optimization, by undergraduate and graduate students for theses and projects as well as by researchers and practitioners from other fields where tools from convex analysis, variational analysis and optimization play a role. All in one, the reviewer warmly recommends this book to anyone interested.? (Sorin-Mihai Grad, zbMATH 1506.90001, 2023)

?This outstanding book will certainly be useful to anyone interested to learn convex analysis, in particular to graduate students and researchers in the field. Most parts of it can also serve as the basis of advanced courses on a variety of topics. In view of the excellence of this first volume, one can expect the best of the announced second one, which will deal with applications of convex analysis.? (Juan Enrique Martínez-Legaz, Mathematical Reviews, February, 2023)

?Every chapter of the book has one section of exercises and one section of commentaries. These sections provide the reader with a lot of information and give him/her great benefits in self-learning. ? The book under review has many things to offer and, surely, it will play an important role in the development of convex analysis ? . The book is very useful for theoretical research and practical use. Thanks to the art of writing of the authors ? .? (Nguyen Dong Yen, Journal of Global Optimization, Vol. 85, 2023)

Table of Contents:

Fundamentals.- Basic theory of convexity.- Convex generalized differentiation.- Enhanced calculus and fenchel duality.- Variational techniques and further subgradient study.- Miscellaneous topics on convexity.- Convexified Lipschitzian analysis.- List of Figures.- Glossary of Notation and Acronyms.- Subject Index.