Overview
- Presents a unified theory of convex analysis, optimization, and related topics in both finite and infinite dimensions
- Highlights a variational approach to convex analysis and applications
- Includes in-text exercises and figures suitable for graduate and undergraduate coursework
Part of the book series: Springer Series in Operations Research and Financial Engineering (ORFE)
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About this book
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.
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.
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Keywords
Table of contents (7 chapters)
Reviews
“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)
“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)
Authors and Affiliations
About the authors
Boris Mordukhovich is Distinguished University Professor of Mathematics at Wayne State University. He has more than 500 publications including several monographs. Among his best known achievements are the introduction and development of powerful constructions of generalized differentiation and their applications to broad classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields. Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from six universities over the world. He is a Highly Cited Researcher in Mathematics. His research has been supported by continued grants from the National Science Foundations and the Air Force Office of Scientific Research.
Nguyen Mau Nam is a Professor of Mathematics at Portland State University. He has published more than 55 research articles and one book in convex analysis withapplications to optimization theory and numerical algorithms. He has received several awards for his research including a best paper award by Journal of Global Optimization in 2021 and the Columbia-Willamette Chapter of Sigma Xi Outstanding Researcher Award in Mathematical Sciences in 2015. His research was supported by grants from the National Science Foundation, the Simons Foundation, and Portland State University.
Nguyen Mau Nam is a Professor of Mathematics at Portland State University. He has published more than 55 research articles and one book in convex analysis withapplications to optimization theory and numerical algorithms. He has received several awards for his research including a best paper award by Journal of Global Optimization in 2021 and the Columbia-Willamette Chapter of Sigma Xi Outstanding Researcher Award in Mathematical Sciences in 2015. His research was supported by grants from the National Science Foundation, the Simons Foundation, and Portland State University.
Bibliographic Information
Book Title: Convex Analysis and Beyond
Book Subtitle: Volume I: Basic Theory
Authors: Boris S. Mordukhovich, Nguyen Mau Nam
Series Title: Springer Series in Operations Research and Financial Engineering
DOI: https://doi.org/10.1007/978-3-030-94785-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-94784-2Published: 25 April 2022
Softcover ISBN: 978-3-030-94787-3Published: 25 April 2023
eBook ISBN: 978-3-030-94785-9Published: 24 April 2022
Series ISSN: 1431-8598
Series E-ISSN: 2197-1773
Edition Number: 1
Number of Pages: XVII, 585
Number of Illustrations: 7 b/w illustrations, 40 illustrations in colour
Topics: Optimization, Numerical Analysis