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On the Density of Positive Proper Efficient Points in a Normed Space

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  • K. F. Ng

    (Chinese University of Hong Kong, Shatin)

  • X. Y. Zheng

    (Chinese University of Hong Kong, Shatin)

Abstract

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S +, we establish some density results of positive weak* efficient elements of A in E(A, S +).

Suggested Citation

  • K. F. Ng & X. Y. Zheng, 2003. "On the Density of Positive Proper Efficient Points in a Normed Space," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 105-122, October.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:1:d:10.1023_b:jota.0000005043.39887.76
    DOI: 10.1023/B:JOTA.0000005043.39887.76
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    References listed on IDEAS

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    1. W. Song, 1997. "Generalization of the Arrow–Barankin–Blackwell Theorem in a Dual Space Setting," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 225-230, October.
    2. X. Y. Zheng, 1998. "Generalizations of a Theorem of Arrow, Barankin, and Blackwell in Topological Vector Spaces," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 221-233, January.
    3. Majumdar, Mukul, 1972. "Some general theorems on efficiency prices with an infinite-dimensional commodity space," Journal of Economic Theory, Elsevier, vol. 5(1), pages 1-13, August.
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    Cited by:

    1. X. Y. Zheng & X. M. Yang & K. L. Teo, 2007. "Some Geometrical Aspects of Efficient Points in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 275-288, May.

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