Strong and Converse Fenchel Duality for Vector Optimization Problems in Locally Convex Spaces

In relation to the vector optimization problem v-minx∈X(f + g ◦ A)(x), with f, g proper and cone-convex functions and A : X → Y a linear continuous operator between separated locally convex spaces, we define a general vector Fenchel-type dual problem. For the primal-dual pair we prove weak, and under appropriate regularity conditions, strong and converse duality. In the particular case when the image space is R we compare the new dual with two other duals, whose definitions were inspired from [9] and [10], respectively. The sets of Pareto efficient elements of the image sets of their feasible sets through the corresponding objective functions prove to be equal, despite the fact that among the image sets of the problems, strict inclusion usually holds. This equality allows us to derive weak, strong and converse duality results for the later two dual problems, from the corresponding results of the first mentioned one. Our results could be implemented in various practical areas, since they provide sufficient conditions for the existence of optimal solutions for vector optimization problems defined on very general spaces. They can be used in medical areas, for example in the study of chronical diseases and in oncology. Received by the editors: 04.12.2008. 2000 Mathematics Subject Classification. 49N15, 32C37, 90C29.