Vector Optimization Theory, Applications, and Extensions
This book presents fundamentals and important results of vector optimization in a general setting. The theory developed includes scalarization, existence theorems, a generalized Lagrange multiplier rule and duality results. Applications to vector approxim
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Johannes Jahn
Vector Optimization Theory, Applications, and Extensions Second Edition
Prof. Dr. Johannes Jahn Universität Erlangen-Nürnberg Department Mathematik Martensstraße 3 91058 Erlangen Germany [email protected]
ISBN 978-3-642-17004-1 e-ISBN 978-3-642-17005-8 DOI 10.1007/978-3-642-17005-8 Springer Heidelberg Dordrecht London New York Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Claudia and Martin
Preface
In vector optimization one investigates optimal elements such as minimal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The problem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineering and economics. Vector optimization problems arise, for example, in functional analysis (the Hahn-Banach theorem, the Bishop-Phelps lemma, Ekeland’s variational principle), multiobjective programming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last two decades vector optimization has been extended to problems with set-valued maps. This new field of research, called set optimization, seems to have important applications to variational inequalities and optimization problems with multivalued data. The roots of vector optimization go back to F.Y. Edgeworth (1881) and V. Pareto (1906) who have already given the definition of the standard optimality concept in multiobjective optimization. But in mathematics this branch of optimization has started with the legendary paper of H.W. Kuhn and A.W. Tucker (1951). Since about vii
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Preface
the end of the 1960’s research is intensively made in vector optimization. It is the aim of this book to pre
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