# Bishop-Phelps Cones and Convexity: Applications to Stability of Vector Optimization Problems

Abstract : In this report we study the stability of cone support points $Min(A|{\cal K})$ of a given set $A$ in a topological vector space $Y¸,$ equipped with a closed convex cone ${\cal K}\subset Y¸.$ We prove sufficient conditions for the lower continuity of $Min(A|{\cal K})$ when $A$ is subjected to perturbations (Theorem 2.2, Theorem 2.3). The crucial assumption is that the set $Min(A|{\cal K})$ is dense in the set of strict cone support points (Definition 2.1). In normed vector spaces $Y$ the set of strict support points contains the set of super efficient points in the sense of Borwein and Zhuang. By making use of the density result for super efficient points Theorem 4.2 gives sufficient conditions for the lower continuity of cone support points for cones with weakly compact bases and the original set $A$ being closed and convex. When ${\cal K}$ is a Bishop-Phelps cone in a Banach space $Y$ we give a simple characterisation of strict support points (Theorem 3.2) which allows us to give a variant of the result of Attouch and Riahi (Theorem 3.4) without any compactness assumption (Theorem 3.5.}
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Submitted on : Wednesday, May 24, 2006 - 1:59:36 PM
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• HAL Id : inria-00073885, version 1

### Citation

Ewa M. Bednarczuk. Bishop-Phelps Cones and Convexity: Applications to Stability of Vector Optimization Problems. [Research Report] RR-2806, INRIA. 1996, pp.11. ⟨inria-00073885⟩

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