Synopses & Reviews
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 leads mathematicians to study questions about 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 for the resolution of these types of problems. These notions have been used largely and traditionally in convex analysis, yet these concepts 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 for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.
Review
From the reviews: "The main purpose of this book is to provide a systematic study of asymptotic cones and asymptotic functions in finite dimensional normed spaces. ... Every chapter ends with bibliographical notes. ... The book is addressed to graduate students at an advanced level and to researchers and practitioners in the fields of optimization theory, nonlinear programming and applied mathematical sciences. ... We recommend this book to all those who are interested in asymptotic analysis and its use." (Constantin Zalinescu, Zentralblatt MATH, Vol. 1017, 2003)
Review
From the reviews:
"The main purpose of this book is to provide a systematic study of asymptotic cones and asymptotic functions in finite dimensional normed spaces. ... Every chapter ends with bibliographical notes. ... The book is addressed to graduate students at an advanced level and to researchers and practitioners in the fields of optimization theory, nonlinear programming and applied mathematical sciences. ... We recommend this book to all those who are interested in asymptotic analysis and its use." (Constantin Zalinescu, Zentralblatt MATH, Vol. 1017, 2003)
Synopsis
This systematic and comprehensive account of asymptotic sets and functions develops a broad and useful theory in the areas of optimization and variational inequalities. The central focus is on problems of handling unbounded situations, using solutions of a given problem in these classes, when for example standard compacity hypothesis is not present. This book will interest advanced graduate students, researchers, and practitioners of optimization theory, nonlinear programming, and applied mathematics.
Table of 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