Directionally generalized differentiation for multifunctions and applications to set-valued programming problems
The aim of the paper is to establish directional optimality conditions for (local) optimal solutions of the set-valued optimization problem \[ \min F(x) \text{ subject to } 0 \in G(x), \,\,x \in \Omega, \] where \(F\) and \(G\) are multifunctions between Asplund spaces. The authors first introduce the definitions of a variety of tangent/contingent/normal cones, (directional) (partial) sequential normal compactness and normal/directionally normal coderivatives and subdifferentials for multifunctions. Using these definitions, they provide calculus rules for directionally gerneralized differentials, namely, first, an intersection rule for directionally limiting normal cones of sets, second, a sum rule for the directionally normal coderivatives of multifunctions and third, calculus rules for the directionally partial subdifferentials of smooth functions. The main results are directional necessary optimality conditions and a normal necessary optimality condition for local minimizers of the set optimization problem \[ \min F(x) \text{ subject to } x \in \Omega. \] Finally, a directionally necessary optimality condition for the original problem with constraint \(0 \in G(x)\) is proved.
scientific article; zbMATH DE number 7032555
scientific article