Asymptotic Cones and Functions in Optimization and Variational Inequalities

Alfred Auslender and Marc Teboulle Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. 264 pp. ISBN 0-387-95520-8 ------------------ 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. A variety of motivations lead mathematicians to study questions revolving around attainment of the infimum in a minimization problem and its stability, duality and minmax theorems, convexification of sets and functions, and maximal monotone maps. For each there is the central problem of handling unbounded situations. Such problems arise in theory but also within the development of numerical methods. The book focuses on the notions of asymptotic cones and associated asymptotic functions that provide a natural and unifying framework to resolve these types of problems. These notions have been used largely and traditionally in convex analysis, yet these concepts also play a prominent and independent role in both convex and nonconvex analysis. This book covers convex and nonconvex problems, offering detailed analysis and techniques that go beyond traditional approaches. The book will serve as a useful reference and self-contained text to researchers, and graduate students in the fields of modern optimization theory and nonlinear analysis.
Contents: Convex Analysis and Set-Valued Maps: A Review--Asymptotic Cones and Functions--Existence and Stability in Optimization Problems--Minimizing and Stationary Sequences--Duality in Optimization Problems--Maximal Monotone Maps and Variational Inequalities.


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