Ebook: Asymptotic Cones and Functions in Optimization and Variational Inequalities

Nonlinear applied analysis and in particular the related ?elds of continuous optimization and variational inequality problems have gone through major developments over the last three decades and have reached maturity. A pivotal role in these developments has been played by convex analysis, a rich area covering a broad range of problems in mathematical sciences and its applications. Separation of convex sets and the Legendre–Fenchel conjugate transforms are fundamental notions that have laid the ground for these fruitful developments. Two other fundamental notions that have contributed to making convex analysis a powerful analytical tool and that haveoftenbeenhiddeninthesedevelopmentsarethenotionsofasymptotic sets and functions. The purpose of this book is to provide a systematic and comprehensive account of asymptotic sets and functions, from which a broad and u- ful theory emerges in the areas of optimization and variational inequa- ties. There is a variety of motivations that led mathematicians to study questions revolving around attaintment of the in?mum in a minimization problem and its stability, duality and minmax theorems, convexi?cation of sets and functions, and maximal monotone maps. In all these topics we are faced with the central problem of handling unbounded situations.

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This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. Mainly it is concerned with the existence of solutions of a given problem in these classes, whenever for example standard compacity hypothesis is not present. Thus it addresses the central problem of handling unbounded situations. This book will be useful to advanced graduate students, researchers, and practitioners in the fields of optimization theory, nonlinear programming, and applied mathematical sciences.

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This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. Mainly it is concerned with the existence of solutions of a given problem in these classes, whenever for example standard compacity hypothesis is not present. Thus it addresses the central problem of handling unbounded situations. This book will be useful to advanced graduate students, researchers, and practitioners in the fields of optimization theory, nonlinear programming, and applied mathematical sciences.
Content:
Front Matter....Pages i-xii
Convex Analysis and Set-Valued Maps: A Review....Pages 1-24
Asymptotic Cones and Functions....Pages 25-80
Existence and Stability in Optimization Problems....Pages 81-118
Minimizing and Stationary Sequences....Pages 119-144
Duality in Optimization Problems....Pages 145-182
Maximal Monotone Maps and Variational Inequalities....Pages 183-232
Back Matter....Pages 233-249

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This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. Mainly it is concerned with the existence of solutions of a given problem in these classes, whenever for example standard compacity hypothesis is not present. Thus it addresses the central problem of handling unbounded situations. This book will be useful to advanced graduate students, researchers, and practitioners in the fields of optimization theory, nonlinear programming, and applied mathematical sciences.
Content:
Front Matter....Pages i-xii
Convex Analysis and Set-Valued Maps: A Review....Pages 1-24
Asymptotic Cones and Functions....Pages 25-80
Existence and Stability in Optimization Problems....Pages 81-118
Minimizing and Stationary Sequences....Pages 119-144
Duality in Optimization Problems....Pages 145-182
Maximal Monotone Maps and Variational Inequalities....Pages 183-232
Back Matter....Pages 233-249
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