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Generalization of the Arrow–Barankin–Blackwell Theorem in a Dual Space Setting

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  • W. Song

    (Harbin Normal University)

Abstract

In this note, we provide general sufficient conditions under which, if F is a compact [resp. w*-compact] subset of the topological dual Y* of a nonreflexive normed space Y partially ordered by a closed convex pointed cone K, then the set of points in F that can be supported by strictly positive elements in the canonical embedding of Y in Y** is norm dense [resp. w*-dense] in the efficient [maximal] point set of F. This result gives an affirmative answer to the conjecture proposed by Gallagher (Ref. 19), and also generalizes the results stated in Ref. 19 and some space specific results given in Refs. 17, 18, and 11.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:95:y:1997:i:1:d:10.1023_a:1022647831434
    DOI: 10.1023/A:1022647831434
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    References listed on IDEAS

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    1. Jonathan M. Borwein, 1983. "On the Existence of Pareto Efficient Points," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 64-73, February.
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    Cited by:

    1. X. D. H. Truong, 2001. "Existence and Density Results for Proper Efficiency in Cone Compact Sets," Journal of Optimization Theory and Applications, Springer, vol. 111(1), pages 173-194, October.
    2. 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.

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