Theory of Vector Optimization
These notes grew out of a series of lectures given by the author at the Univer sity of Budapest during 1985-1986. Additional results have been included which were obtained while the author was at the University of Erlangen-Niirnberg under a grant of the
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Theory of Vector Optimization
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Editorial Board H. Albach M. Beckmann (Managing Editor) P.Ohrymes G. Fandei G. Feichtinger J. Green W. Hildenbrand W. Krelle (Managing Editor) H.P. Künzi K. Ritter R. Sato U. Schittko P. Schönfeld R. Selten Managing Editors Prof. Dr. M. Beckmann Brown University Providence, RI 02912, USA Prof. Or. W. Krelle Institut für Gesellschafts- und Wirtschaftswissenschaften der Universität Bonn Adenauerallee 24-42, 0-5300 Bonn, FRG Author Or. Oinh The Luc Institute of Mathematics P.Box 631 Boho 10000 Hanoi, Vietnam
ISBN 978-3-540-50541-9 ISBN 978-3-642-50280-4 (eBook) DOI 10.1007/978-3-642-50280-4 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© SprinQer-VerlaQ Berlin HeidelberQ 1989 2142/3140-543210
To My Mother and the Memory of My Father
Preface These notes grew out of a series of lectures given by the author at the University of Budapest during 1985-1986. Additional results have been included which were obtained while the author was at the University of Erlangen-Niirnberg under a grant of the Alexander von Humboldt Foundation. Vector optimization has two main sources coming from economic equilibrium and welfare theories of Edgeworth (1881) and Pareto (1906) and from mathematical backgrounds of ordered spaces of Cantor (1897) and Hausdorff (1906). Later, game theory of Borel (1921) and von Neumann (1926) and production theory of Koopmans (1951) have also contributed to this area. However, only in the fifties, after the publication of Kuhn-Tucker's paper (1951) on the necessary and sufficient conditions for efficiency, and of Deubreu's paper (1954) on valuation equilibrium and Pareto optimum, has vector optimization been recognized as a mathematical discipline. The stretching development of this field began later in the seventies and eighties. Today there are a number of books on vector optimization. Most of them are concerned with the methodology and the applications. Few of them offer a systematic study of the theoretical aspects.The aim of these notes is to provide a unified background of vector optimization,with the emphasis on nonconvex problems in infinite dimensional spaces ordered by convex cones. The notes are arranged into six chapters. The first chapter presents preliminary material. It contains a study of nonconvex analysis with respect to convex cones. In chapter 2, we introduce main concepts of vector optimization such as preference order, efficiencies, vector optimality etc.
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