Vector Optimization with Infimum and Supremum
The theory of Vector Optimization is developed by a systematic usage of infimum and supremum. In order to get existence and appropriate properties of the infimum, the image space of the vector optimization problem is embedded into a larger space, which is
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Vector Optimization The series in Vector Optimization contains publications in various fields of optimization with vector-valued objective functions, such as multiobjective optimization, multi criteria decision making, set optimization, vector-valued game theory and border areas to financial mathematics, biosystems, semidefinite programming and multiobjective control theory. Studies of continuous, discrete, combinatorial and stochastic multiobjective models in interesting fields of operations research are also included. The series covers mathematical theory, methods and applications in economics and engineering. These publications being written in English are primarily monographs and multiple author works containing current advances in these fields.
Andreas Löhne
Vector Optimization with Infimum and Supremum
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Andreas Löhne Martin-Luther-Universität Halle-Wittenberg NWF II - Institut für Mathematik Theodor-Lieser-Str. 5 06120 Halle Germany [email protected]
ISSN 1867-8971 ISBN 978-3-642-18350-8 DOI 10.1007/978-3-642-18351-5
e-ISSN 1867-898X e-ISBN 978-3-642-18351-5
Springer Heidelberg Dordrecht London Newyork
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To Jana and Pascal
Preface
Infimum and supremum are indispensable concepts in optimization. Nevertheless their role in vector optimization has been rather marginal. This seems to be due the fact that their existence in partially ordered vector spaces is connected with restrictive assumptions. The key to an approach to vector optimization based on infimum and supremum is to consider set-valued objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions. These ideas lead to a novel exposition of vector optimization. The reader is not only required to familiarize with several new concepts, but also a change of philosophy is suggested to those being acquainted with the classical approaches. The goal of this monograph is to cover
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