锥的分离定理及其应用

A Cone Separation Theorem and Its Application

本文给出了一个雄的分离定理并利用它给出了在Banach空间情形Benson真有效点的一个刻划．

This paper gives a cone separationt theorem and a characterization of Benson's proper efficiency in a Banach space. The results are as follows: Theorem 1. Let E be a Banach space and P E a closed convex cone satisfying property (K), and Q E a closed convex cone with P ∩ Q = {θ}. Then there edests a equence of cones { Pj} satisfying weak property (π) which separate strictly P and Q. Pi has the following additional properties: (i) P Pj,+l (ii) intp (iii) Remark 1. In theorem 1, if we assume that P satisfies weak propeds (π) and Q is a LWC cone, then the same result is still true.Let E be a Banach space, P=E a closed convex cone and YE a set. y∈E is called an efficiency of Y w. r. t. P if ( Y-y) ∩ (- p) = {θ}. All such points are denoted by E(Y |P).Definition 1. y∈ E(Y | P) is called a Bensons,s proper efficiency if cone(y+p-y) ∩ (-p) = {θ}.Definition 2. y is called a global effciency, if y∈ E(Y|P) where P is a convex cone such that P\{θ} nit .Theorem 2. If P satisfies property (π), then the above two defirutions are equivalent.