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Recent Advances and Historical Development of Vector Optimization pp 114–128Cite as

Sets of Efficiency in a Normed Space and Inner Product

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 294))

Abstract

In a normed space X the distances to the points of a given set A being considered as the objective functions of a multicriteria optimization problem, we define four sets of efficiency (efficient, strictly efficient, weakly efficient and properly efficient points). Instead of studying properties of the sets of efficiency according to properties of the norm, we investigate an inverse problem: deduce properties of the norm of X from properties of the sets of efficiency, valid for every finite subset A of X.

We thus obtain some new characterizations of inner product spaces. The tools are classical James’ characterizations of inner product spaces and a description of each set of efficiency which looks like the description of the convex hull of A.

The same machinery allows to study polyhedral spaces where only partial results are obtained. Some characterizations of strictly convex spaces are also given.

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Author information

Authors and Affiliations

  1. Laboratoire d’Analyse Numérique, Université de Bourgogne, B.P. 138, 21004, Dijon Cedex, France

    Roland Durier

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  1. Roland Durier
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Editor information

Editors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, D-8520, Erlangen, Germany

    Johannes Jahn

  2. Fachbereich Mathematik, Technische Hochschule Darmstadt, Schloßgartenstr. 7, D-6100, Darmstadt, Germany

    Werner Krabs

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© 1987 Springer-Verlag Berlin Heidelberg

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Durier, R. (1987). Sets of Efficiency in a Normed Space and Inner Product. In: Jahn, J., Krabs, W. (eds) Recent Advances and Historical Development of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46618-2_6

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  • DOI: https://doi.org/10.1007/978-3-642-46618-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18215-3

  • Online ISBN: 978-3-642-46618-2

  • eBook Packages: Springer Book Archive

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