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This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.
Preface.- Notation, Terminology and Some Functional Analysis.- Basics in Optimization.- Continuity of Set-valued Mappings.- Lipschitz Continuity of Polyhedral Mappings.- Metric Regularity.- Lyusternik-Graves Theorem.- Mappings with Convex Graphs.- Derivative Criteria for Metric Regularity.- Strong Regularity.- Variational Inequalities over Polyhedral Sets.- Nonsmooth Inverse Function Theorems.- Lipschitz Stability in Optimization.- Strong Subregularity.- Continuous Selections.- Radius of Regularity.- Newton Method for Generalized Equations.- The Constrained Linear-Quadratic Optimal Control Problem.- Regularity in Nonlinear Control.- Discrete Approximations.- Optimal Feedback Control.- Model Predictive Control.- Bibliographical Remarks and Further Reading.
Description
This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.
