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Implicit Functions and Solution Mappings : A View from Variational Analysis / by Asen L. Dontchev, R. Tyrrell Rockafellar
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Monographs in Mathematics | SpringerLink BücherPublisher: New York, NY : Springer-Verlag New York, 2009Description: Online-Ressource (XII, 376p. 12 illus, digital)ISBN:- 9780387878218
- 515 23
- 515.8 22
- QA331.5
Contents:
Summary: Functions Defined Implicitly by Equations -- Implicit Function Theorems for Variational Problems -- Regularity properties of set-valued solution mappings -- Regularity Properties Through Generalized Derivatives -- Regularity in infinite dimensions -- Applications in Numerical Variational Analysis.Summary: The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This book treats the implicit function paradigm in the classical framework and beyond, focusing largely on properties of solution mappings of variational problems. The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided. This valuable book is a major achievement and is sure to become a standard reference on the topic.PPN: PPN: 1649893736Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""CONTENTS""; ""Preface""; ""Acknowledgements""; ""Chapter 1. Functions defined implicitly by equations""; ""1A. The classical inverse function theorem""; ""1B. The classical implicit function theorem""; ""1C. Calmness""; ""1D. Lipschitz continuity""; ""1E. Lipschitz invertibility from approximations""; ""1F. Selections of multi-valued inverses""; ""1G. Selections from nonstrict differentiability""; ""Chapter 2. Implicit function theorems for variational problems""; ""2A. Generalized equations and variational problems""; ""2B. Implicit function theorems for generalized equations""
""2C. Ample parameterization and parametric robustness""""2D. Semidifferentiable functions""; ""2E. Variational inequalities with polyhedral convexity""; ""2F. Variational inequalities with monotonicity""; ""2G. Consequences for optimization""; ""Chapter 3. Regularity properties of set-valued solution mappings""; ""3A. Set convergence""; ""3B. Continuity of set-valued mappings""; ""3C. Lipschitz continuity of set-valued mappings""; ""3D. Outer Lipschitz continuity""; ""3E. Aubin property, metric regularity and linear openness""; ""3F. Implicit mapping theorems with metric regularity""
""3G. Strong metric regularity""""3H. Calmness and metric subregularity""; ""3I. Strong metric subregularity""; ""Chapter 4. Regularity properties through generalized derivatives""; ""4A. Graphical differentiation""; ""4B. Derivative criteria for the Aubin property""; ""4C. Characterization of strong metric subregularity""; ""4D. Applications to parameterized constraint systems""; ""4E. Isolated calmness for variational inequalities""; ""4F. Single-valued localizations for variational inequalities""; ""4G. Special nonsmooth inverse function theorems""; ""4H. Results utilizing coderivatives""
""Chapter 5. Regularity in infinite dimensions""""5A. Openness and positively homogeneous mappings""; ""5B. Mappings with closed convex graphs""; ""5C. Sublinear mappings""; ""5D. The theorems of Lyusternik and Graves""; ""5E. Metric regularity in metric spaces""; ""5F. Strong metric regularity and implicit function theorems""; ""5G. The Bartle�Graves theorem and extensions""; ""Chapter 6. Applications in numerical variational analysis""; ""6A. Radius theorems and conditioning""; ""6B. Constraints and feasibility""; ""6C. Iterative processes for generalized equations""
""6D. An implicit function theorem for Newton�s iteration""""6E. Galerkin�s method for quadratic minimization""; ""6F. Approximations in optimal control""; ""References""; ""Notation""; ""Index""
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