Asymptotic Cones and Functions in Optimization and
Alfred Auslender and Marc Teboulle
Springer Monographs in Mathematics,
Springer-Verlag, New York, 2003.
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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