Global and local optimality conditions in set-valued optimization problems

Through the paper, X and Y are normed vector spaces; however, most of the results remain true in the more general setting of locally convex spaces. We denote by X∗ and Y∗ the topological dual spaces of X and Y . We consider a pointed closed convex cone Q ⊂ Y which introduces a partial order on Y by the equivalence y1 ≤Q y2 ⇔ y2 − y1 ∈Q; we also suppose, in general, that Q has nonempty interior. We set Q+ := {y∗ ∈ Y∗ | y∗(y) ≥ 0, ∀y ∈Q} for the dual cone of Q and Q+i := {y∗ ∈ Y∗ | y∗(y) > 0, ∀y ∈Q \ {0}} for the quasi-interior of Q+. We take a set-valued map F from X into Y . As usual, we denote the graph and domain of F, respectively, by