Variational Analysis and Generalized Differentiation II

Variational analysis has been recognized as a fruitful area in mathematics that on the one hand deals with the study of optimization and equilibrium problems and on the other hand applies optimization, perturbation, and approximation ideas to the analysis of a broad range of problems that may not be of a variational natur. One of the most characteristic features of modern variational analysis is the intrinsic presence of nonsmoothness, which naturally enters not only through initial data of optimization-related problems but largely via variational principles and perturbation techniques. Thus generalized differential lies at the hear of variational analysis and its applications.

This monographs contains a comprehensive and and state-of-the art study of the basic concepts and principles of variational analysis and generalized differentiation in both finite-dimensional and infinite dimensional spaces and presents numerous applications to problems in the optimization, equilibria, stability and sensitivity, control theory, economics, mechanics, etc. The book is published in two volumes, the first of which is mainly devoted to the basic theory of variational analysis and generalized differentiations, while the second volume contains various applications. Both volumes contain abundant bibliographies and extensive commentaries.

This book will be of interest to researchers and graduate students in mathematical sciences. It may also be useful to a broad range of researchers, practitioners, and graduate students involved in the study and applications of variational methods in economics, engineering, control systems, operations research, statistics, mechanics, and other applied sciences.

• differential inclusions
• economics
• equilibria
• evolution systems
• multi-objective optimization
• optimal control
• optimization
• partial differential equations

From the reviews:

"The combination of the two volumes provide a thorough, self-contained, well-motivated view of an important line of research in variational analysis. This is a reference text which readers having an interest in variational analysis or any of the application areas should find to be a very useful resource." (Adam B. Levy, SIAM Review, Vol. 48 (3), 2006)

"This two-volume book is a deep and wide treatise on variational analysis and its relationships with several fields of mathematics … . Among several aspects which make the book beautiful and interesting, there is the fact that the reader is met at a classical level of mathematical analysis … . it is precious for all scholars who want to learn modern variational analysis and related fields; indeed, this is a fundamental tool for those who are involved in the applications of mathematics." (Franco Giannessi, Journal of Optimization Theory and Applications, Vol. 131 (2), 2006)

"This … monograph reflects the abiding and enthusiastic interest of the author in variational analysis. … Besides mathematics, there are extensive commentaries on the history and genesis of major topics under consideration … open problems, and an extensive bibliography (1379 references). These features make the book appreciated by students and experts in the field. The book will definitely serve as a fundamental read and also a useful reference. It is highly recommended to all libraries and researchers in the area of variational analysis and optimization." (Panos Pardalos, Journal of Global Optimization, Vol. 36, 2006)

"The new two-volume monograph by B.S. Mordukhovich contains a comprehensive presentation of … calculus together with numerous applications and extensive commentaries. … The second volume dedicated to applications, keeps the pace with the first volume as regards the systematic and rigorous exposition and high theoretical level. … this outstanding work contains aconsiderable number of previously unpublished results. We believe that its both volumes will soon become an indispensable part of the library of each researcher interested in the fields listed above." (Jirí V. Outrata and Michal Cervinka, Mathematical Methods of Operations Research, Vol. 65 (1), 2007)

"The book under review offers an original and powerful way to understand the behavior of sets and functions which are neither smooth nor convex. … The author includes an extensive bibliography of 1379 references … . The book is well written in a self-contained way, in the sense that most topics are developed from their beginnings. … Thus the book should prove to be, for beginners and specialists, a key reference for variational or nonsmooth analysis in both finite-dimensional spaces and infinite-dimensional Asplund spaces." (Lionel Thibault, Mathematical Reviews, Issue 2007 b)

"The author succeeded to place at our disposal a comprehensive and deep insight into a fruitful and continuously developing area in mathematics. He presented the theory of modern variational analysis and generalized differentiation in its full generality. This allowed to cover a large range of applications. We appreciate that the book synthetize a life time work of an important mathematician, and warmly recommend it to specialists in mathematical analysis, differential equations or mathematical economics." (Mihai Pascu, Revue Roumaine de Mathématique Pures et Appliquées, Vol. LII (5), 2007)

“The present monograph provides a comprehensive theory of generalized differentiation and its applications in infinite dimensions. … Each chapter … is complemented by historic and bibliographic remarks which are based on an impressive list of references and which allow to put parallel developments in nonsmooth analysis into a context with the author’s work. … will become a fundamental reference in variational analysis and that they will stimulate a lot of fruitfulapplications in the abundance of problems where nonsmoothness is an inherent issue.” (René Henrion, Operations Research Letters, Vol. 35 (4), July, 2007)

• College of Science, Department of Mathematics, Wayne State University, Detroit, USA

  Boris S. Mordukhovich

Ph.D. in Mathematics (1973); distinguished faculty and lifetime scholar of the Academy of Scholars; more than 200 publications (including books and patents); many outstanding research and teaching awards; numerous invited talks to various meetings (e.g., 15 keynote presentations during the last year); organizer of conference, special sessions, and issues of journals; Editorial Boards of 11 international journals; grants and awards from NSF, NATO, NSERC, BSF, Australian Research Council, etc.

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