An Optimal Alternative Theorem and Applications to Mathematical Programming

Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set A ⊂ 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 characterize the two-dimensionality in terms of the validity of an alternative theorem. 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.