Journal of Optimization Theory and Applications

, Volume 91, Issue 2, pp 419–437 | Cite as

Density theorems for generalized Henig proper efficiency

  • E. K. Makarov
  • N. N. Rachkovski
Contributed Papers

Abstract

We develop a new, simple technique of proof for density theorems (i.e.,for the sufficient conditions to guarantee that the proper efficient points of a set are dense in the efficient frontier) in an ordered topological vector space. The results are the following: (i) the set of proper efficient points of any compact setQ is dense in the set of efficient points with respect to the original topology of the space whenever the ordering coneK is weakly closed and admits strictly positive functionals; moreover, ifK is not weakly closed, then there exists a compact set for which the density statement fails; (ii) ifQ is weakly compact, then we have only weak density, but ifK has a closed bounded base, then we can assert the density with respect to the original topology, (iii) there exists a similar possibility to assert the strong density for weakly compactQ if additional restrictions are placed onQ instead ofK. These three results are obtained in a unified way as corollaries of the same statement. In this paper, we use the concept of proper efficiency due to Henig. We extend his definition to the setting of a Hausdorff topological vector space.

Key Words

Vector optimization positive proper efficiency Henig proper efficiency density theorem 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • E. K. Makarov
    • 1
  • N. N. Rachkovski
    • 1
  1. 1.Institute of MathematicsAcademy of Sciences of BelarusMinskBelarus

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