Bishop-Phelps Cones and Convexity: Applications to Stability of Vector Optimization Problems
Ewa M. Bednarczuk, 54 - Villers-les-Nancy (France). Unite de Recherche de Lorraine Institut National de Recherche en Informatique et en Automatique (INRIA) · 1996
In this report we study the stability of cone support points Min(A|K) of a given set A in a topological vector space Y, equipped with a closed convex cone K \\subset Y. We prove sufficient conditions for the lower continuity of Min(A|K) when A is subjected to perturbations (Theorem 2.2, Theorem 2.3). The crucial assumption is that the set Min(AjK) 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 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 a...