Variational Analysis and Applications

by Boris S. Mordukhovich

Overview

Building on fundamental results in variational analysis, this monograph presents new and recent developments in the field as well as selected applications. Accessible to a broad spectrum of potential readers, the main material is presented in finite-dimensional spaces. Infinite-dimensional developments are discussed at the end of each chapter with comprehensive commentaries which emphasize the essence of major results, track the genesis of ideas, provide historical comments, and illuminate challenging open questions and directions for future research.

The first half of the book (Chapters 1–6) gives a systematic exposition of key concepts and facts, containing basic material as well as some recent and new developments. These first chapters are particularly accessible to masters/doctoral students taking courses in modern optimization, variational analysis, applied analysis, variational inequalities, and variational methods. The reader’s development of skills will be facilitated as they work through each, or a portion of, the multitude of exercises of varying levels. Additionally, the reader may find hints and references to more difficult exercises and are encouraged to receive further inspiration from the gems in chapter commentaries. Chapters 7–10 focus on recent results and applications of variational analysis to advanced problems in modern optimization theory, including its hierarchical and multiobjective aspects, as well as microeconomics, and related areas. It will be of great use to researchers and professionals in applied and behavioral sciences and engineering.

Product Details

ISBN-13:	9783319927732
Publisher:	Springer International Publishing
Publication date:	08/03/2018
Series:	Springer Monographs in Mathematics
Edition description:	1st ed. 2018
Pages:	622
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Boris S. Mordukhovich is a Distinguished University Professor of Mathematics at Wayne State University. His areas of research include optimization and variational analysis, control theory, nonlinear systems, generalized differentiation, and various applications. Professor Mordukhovich has authored more than 400 publications including two volumes in Springer’s Grundlehren series entitled “Variational Analysis and Generalized Differentiation.” He is a Fellow of the American Mathematical Society (AMS) and the Society of Industrial and Applied Mathematics (SIAM).

Table of Contents

Preface.- Part 1. Basics of First-Order Analysis.-1. First-Order Generalized Differentiation.- 2. Extremality, Well-Posedness, Calculus Rules.- 3. Optimization via Generalized Differentiation.- Part 2. Second-Order Analysis.- 4. Second-Order Subdifferentials.- 5. Second-Order Subdifferential Calculus.- 6. Full and Tilt Stability of Local Minimizers.- 7. Full Stability for Parametric Variational Systems.- Part 3. Selected Applications.- 8. Full and Tilt Stability in Constrained Optimization.-9. Higher-Order Stability and Numerical Analysis.- 10. Optimal Control of the Sweeping Process.- 11. Microeconomic Modeling.- References.- List of Statements.- Glossary of Notation.- Subject Index.

Editorial Reviews

This exceptionally well-written research monograph … presents material which is of profound importance and is meticulously chosen and organized. The mathematically precise and crisp presentation style would undoubtedly help the reader in acquiring a deep understanding of the subject. I believe that this monograph would not only be of great benefit to the research community actively engaged in variational analysis but would also encourage newcomers who are willing to learn the subject.” (Akhtar A. Khan, SIAM Review, Vol. 62 (1), 2020)

“The book provides a self-contained, systematic and comprehensive exposition of keys concepts and principles of variational analysis, offering to specialists an updated view on the state of the art of a fast-growing field of research … . the rich anthology of exercises, guiding and stimulating the reader's comprehension, makes the book a usable text for teaching variational analysis to a large audience of graduate students, researchers and practitioners (not only in the mathematical sciences).” (Amos Uderzo, Mathematical Reviews, July, 2019)

“I am enthusiastic about its highly reader-friendly, well-structured presentation. It makes the book, apart from its mathematical contents, an aesthetic joy. … I consider this monograph to be a highly instructive, well-structured and clearly written presentation of modern variational analysis from the perspective of nonconvex and dual generalized differential calculus. I do not hesitate to recommend this excellent work to everybody interested in this field of mathematics, to specialists as well as to young researchers.” (René Henrion, Optimization Letters, Optimization Letters, Vol. 13, 2019)

“Written by an expert in the areas of variational analysis and optimization and based on his didactic experience, this is an excellent textbook. By the wealth of information contained in the second part of the book and in exercises,it can be also used by researchers in optimization theory and its applications as a reference text.” (S. Cobzaş, Studia Universitatis Babes-Bolyai Mathematica, Vol. 6 (04). 2018)

“This book is on the one hand a valuable tool for teaching and learning variational analysis, with exercises ranging from medium level to more difficult and numerous illustrative examples and figures. On the other hand, it represents a basis for future research in various fields of mathematics and its applications.” (Radu Ioan Bot, zbMATH 1402.49003, 2019)

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