ESSENTIAL WEAK EFFICIENT SOLUTIONS FOR VECTOR MAXIMIZATION PROBLEMS
G Chen · 1983
This paper deals with a problem about the stability of the vector maximization. A topological structure is introduced to make the family of vector maximization problems a complete metric space. Then the essential weak efficient solution is defined and the vector maximization problems with all their weak efficient solutions being essential are proved to be everywhere dense in the space.Let (X, d) be a totally bounded complete metric space, R~m an m-dimensional Euclidean space, and SR~m an open convex cone. We denote by a complete metric space of all nonempty compact subsets of X with Hausdorff's metric function . We denote by p a vector maximization problem where f is a bounded continuous function from X into R~m, R ∈. We denote by P all p under the above conditions; then P≡C_m[X]×. We denote by M(p) all weak efficient solutions of p∈P.Definition. x∈M (p) is an essential weak efficient solution of p, if corresponding to e0 there exists δ0 such that x∈V(e, M(g)) whenever g∈P and h(p, g)δ.The main results:Lemma 3. The multivalued mapping M is upper semicontinuous on P.Theorem 1. All weak efficient solutions of p∈P are essential weak efficient solutions if and only if p is a point of continuity of M.Theorem 2. For every p∈P and an arbitrary e0, there exists g∈P such that h(p, g)e and that every weak efficient solution of g is an essential weak efficient solution. In other words, the set of all points of continuity of the mapping M is everywhere dense in P.Theorem 3. If the vector maximization problem p has a single weak efficient solution, then this solution is an essential weak efficient solution.