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
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Preface
Nonlinear applied analysis and in particular the related fields 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 have often been hidden in these developments are the notions of asymptotic 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 useful theory emerges in the areas of optimization and variational inequalities. There is a variety of motivations that led mathematicians to study questions revolving around attaintment of the infimum in a minimization problem and its stability, duality and minmax theorems, convexification of sets and functions, and maximal monotone maps. In all these topics we are faced with the central problem of handling unbounded situations. This is particularly true when standard compactness hypotheses are not present. The appropriate concepts and tools needed to study such kinds of problems are vital not only in theory but also within the development of numerical methods. For the latter, we need not only to prove that a sequence generated by a given algorithm is well defined, namely an existence
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result, but also to establish that the produced sequence remains bounded. One can seldom directly apply theorems of classical analysis to answer to such questions. The notions of asymptotic cones and associated asymptotic functions provide a natural and unifying framework to resolve these types of problems. These notions have been used mostly and traditionally in convex analysis, with many results scattered in the literature. Yet these concepts also have a prominent and independent role to play in both convex and nonconvex analysis. This book presents the material reflecting this last point with many parts, including new results and covering convex and nonconvex problems. In particular, our aim is to demonstrate not only the interplay between classical convex-analytic results and the asymptotic machinery, but also the wide potential of the latter in analyzing variational problems. We expect that this book will be useful to graduate students at an advanced level as well as to researchers and practitioners in the fields of optimization theory, nonlinear programming, and applied mathematical sciences. We decided to use a style with detailed and often transparent proofs. This might sometimes bore the more advanced reader, but should at least make the reading of the book easier and hopefully even enjoyable. The material is presented within the finite-dimensional
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