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
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.
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Flores-Bazán, F., Hadjisavvas, N. & Vera, C. An Optimal Alternative Theorem and Applications to Mathematical Programming. J Glob Optim 37, 229–243 (2007). https://doi.org/10.1007/s10898-006-9046-8
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DOI: https://doi.org/10.1007/s10898-006-9046-8