An Optimal Alternative Theorem and Applications to Mathematical Programming

Abstract

Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set $$A \subset Y $$ , we provide various equivalences to the implication $$A \cap (-{\rm int}\,P) = \emptyset \Longrightarrow {\rm co}(A)\cap (-{\rm int}\, P) = \emptyset, $$ 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.