Higher-order Mond-Weir duality for set-valued optimization (Q929605)

Let \(X\) be a Banach space and \(Y\) and \(Z\) be two ordered Banach spaces. Let \(D\) with int\(D \neq \emptyset\) be the pointed closed convex cone which defines the order in \(Z\). Suppose that \(F:X \mapsto 2^{Y}\) and \(G:X \mapsto 2^{Z}\) are two set-valued mappings. The considered vector optimization problem is the following: min \(F(x)\) s.t. \(G(x) \cap \left( -D \right) \neq \emptyset\). The paper provides weak duality, strong duality and converse duality results for such optimization problems with set-valued mappings. By using the mth-order contingent derivative, the authors introduce a kind of higher-order Weir-Mond-type duality, which is presented as a generalization of Weir-Mond duality for single-valued functions presented in \textit{T. Weir} and \textit{B. Mond} [Bull. Austral. Math. Soc. 39, 287--299 (1989; Zbl 0651.90083)].