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Operations-Research-Spektrum

, Volume 16, Issue 4, pp 261–265 | Cite as

Properly efficient solutions for vectorial norm approximation

  • Gert Wanka
Decision Theory

Abstract

In the present paper a general vectorial best approximation problem using vectorial norms with respect to properly efficient solutions is considered. Necessary and sufficient optimality conditions for such solutions are derived. This is done on base of scalarization and studying a corresponding dual problem to the scalar optimization problem. Also optimality conditions in subdifferential form are formulated. For the necessity of the optimality conditions especially convexity assumptions are essential. But under only very weak supposition these conditions are sufficient.

Key words

Vector optimization vectorial approximation optimality conditions properly efficiency duality 

Zusammenfassung

In der Arbeit werden eigentlich effiziente Lösungen für ein allgemeines vektorielles Bestapproximationsproblem betrachtet. Das Approximations-problem ist auf der Basis vektorieller Normen formuliert. Unter Verwendung von Skalarisierung werden notwendige und hinreichende Optimalitätsbedingungen hergeleitet. Dazu wird insbesondere ein skalares Dualproblem konstruiert und es werden entsprechende Dualitätsaussagen angegeben. Weiterhin werden Optimalitätsbedingungen in Subdifferentialform dargestellt. Für die Notwendigkeit der Optimalitätsbedingungen sind insbesondere Konvexitätsvoraussetzungen wesentlich. Unter jedoch sehr schwachen Voraussetzungen sind diese Bedingungen hinreichend.

Schlüselwörter

Vektoroptimierung vektorielle Approximation Optimalitätsbedingungen eigentliche Effizienz Dualität 

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Gert Wanka
    • 1
  1. 1.Fakultät für MathematikTechnische Universität Chemnitz-ZwickauChemnitzGermany

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