Maximal points with respect to cone dominance in Banach spaces and their existence

Abstract

Given a convex cone Λ in a Banach space V, an examination of the cone maximal points of a set X in V (with respect to the cone dominance induced by Λ) with respect to their characterization and existence is undertaken. The totality of cone maximal points of X is called the conical frontier of X. Comparisons of the conical frontiers of related sets and corresponding to related cones are made. By relaxing the compactness requirements of the underlying set X and by assuming some cone-related weaker forms of compactness, existence theorems for cone maximal points are developed. These theorems are believed to be generalizations of the existing results in one way or another. Maximizing points on X of certain linear functionals in the dual cone Λ* of Λ provide natural examples of cone maximal points. Properties characterizing a maximizing point of a linear functional in Λ*, including the generalized version of Geoffrion's characterization of proper efficiency, are compiled and proved to be valid characterizations. Functionals in Λ* with special properties are studied. Existence theorems are also obtained for the maximizing points of these functionals. © 1984 Plenum Publishing Corporation.