Abstract

Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set ⊂ Y , we provide various equivalences to the implication A ∩ (-int P) = ∅ ⇒ co(A)∩ (-int P) = ∅, among them, to the pointedness of cone(A + int P). This allows us to establish an optimal alternative theorem, suitable for vector optimization problems. In addition, we present an optimal alternative theorem which characterizes two-dimensional spaces in the sense that it is valid if, and only if, the space is at most two-dimensional. Applications to characterizing weakly efficient solutions through scalarization; the zero (Lagrangian) duality gap; the Fritz-John optimality conditions for a class of nonconvex nonsmooth minimization problems, are also presented. © Springer Science+Business Media B.V. 2007.