Vector minimization oj a relation F valued in an ordered vector space under a constraint A consists in finding x0 ∈ A, w0 ∈ Fx0 such that w0 is minimal in FA. To a family of vector minimization problems minimize F(x,y), y ∈ Y, one associates a Lagrange relation L(x,ξ,y0) = ∪y∈Yx∈X (F(x,y) − ξ(y) + ξ(y0)) where ξ belongs to an arbitrary class Ξ of mappings. For this type of problems, there exist several notions of solutions. Some useful characterizations of existential solutions are established and, consequently, some necessary conditions of optimality are derived. One result of intermediate duality is proved with the aid of the scalarization theory. Existence theorems for existential solutions are given and a comparison of several exact duality schemes is established, more precisely in the convex case it is shown that the majority of exact duality schemes can be obtained from one result of S. Dolecki and C. Malivert.
