Journal of Optimization Theory and Applications

, Volume 71, Issue 3, pp 613–620 | Cite as

Bases of convex cones and Borwein's proper efficiency

  • D. M. Zhuang
Technical Note

Abstract

In this note, we establish some interesting relationships between the existence of Borwein's proper efficient points and the existence of bases for convex ordering cones in normed linear spaces. We show that, if the closed unit ball in a smooth normed space ordered by a convex cone possesses a proper efficient point in the sense of Borwein, then the ordering cone is based. In particular, a convex ordering cone in a reflexive space is based if the closed unit ball possesses a proper efficient point. Conversely, we show that, in any ordered normed space, if the ordering cone has a base, then every weakly compact set possesses a proper efficient point.

Key Words

Efficiency proper efficiency Borwein's proper efficiency convex cones bases of cones smooth normed spaces 

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Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • D. M. Zhuang
    • 1
  1. 1.Department of Mathematics and Computer StudiesMount St. Vincent UniversityHalifaxCanada

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