Rockafellar R.T., Wets R.J.B. Variational Analysis

Corr. 3rd. printing. — Heidelberg: Springer, 2010. — 743 p. — (A Series of Comprehensive Studies in Mathematics, 317). — ISBN: 3642083048.In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. Rather than detailing all the different approaches that researchers have been occupied with over the years in the search for the right ideas, we seek to reduce the general theory to its key ingredients as now understood, so as to make it accessible to a much wider circle of potential users. But within that consolidation, we furnish a thorough and tightly coordinated exposition of facts and concepts. Several books have already dealt with major components of the subject. Some have concentrated on convexity and kindred developments in realms of nonconvexity. Others have concentrated on tangent vectors and subderivatives more or less to the exclusion of normal vectors and subgradients, or vice versa, or have focused on topological questions without getting into generalized differentiability. Here, by contrast, we cover set convergence and set-valued mappings to a degree previously unavailable and integrate those notions with both sides of variational geometry and subdifferential calculus. We furnish a needed update in a field that has undergone many changes, even in outlook. In addition, we include topics such as maximal monotone mappings, generalized second derivatives, and measurable selections and integrands, which have not in the past received close attention in a text of this scope. (For lack of space, we say little about the general theory of critical points, although we see that as a close neighbor to variational analysis). Because of the large volume of material and the challenge of unifying it properly, we had to [b"draw the line"[/b] somewhere. We chose to keep to finite-dimensional spaces so as not to cloud the picture with the many complications that a treatment of infinite-dimensional spaces would bring.
We envision that this book will be useful to graduate students, researchers and practitioners in a range of mathematical sciences, including some front-line areas of engineering and statistics that draw on optimization. We have aimed at making available a handy reference for numerous facts and ideas that cannot be found elsewhere except in technical papers, where the lack of a coordinated terminology and notation is currently a formidable barrier. At the same time, we have attempted to write this book so that it is helpful to readers who want to learn the field, or various aspects of it, step by step. We have provided many figures and examples, along with exercises accompanied by guides.Max and Min.
Convexity.
Cones and Cosmic Closure.
Set Convergence.
Set-Valued Mappings.
Variational Geometry.
Epigraphical Limits.
Subderivatives and Subgradients.
Lipschitzian Properties.
Subdifferential Calculus.
Dualization.
Monotone Mappings.
Second-Order Theory.
Measurability.Index of Statements.
Index of Notation.
Index of Topics.True PDF