On Cone-Efficiency, Cone-Convexity and Cone-Compactness
Roger Hartley · SIAM Journal on Applied Mathematics · 1978
The properties of efficient (admissible) points of subsets of $R^n $ are discussed in the case when the space is ordered by a convex cone. It is demonstrated that the notion of cone-compactness (a generalization of compactness) is sufficient to guarantee the existence of an efficient point. The relationship between the set of efficient points and the optimal sets of certain linear functions is elucidated in a series of theorems extending previous results, when the underlying set satisfies a weakened convexity assumption (cone-convexity). Geoffrion’s concept of proper efficiency is generalized and after showing that our definition is a valid generalization, the appropriate characterization theorem is established.