Directionally generalized differentiation and applications to set-valued optimization, multiobjective optimal control problems

In this paper, we study a set-valued mathematical programming problem and a multiobjective discrete-time optimal control problem, via the directionally generalized differentiation. By establishing scalarization formulas for the directional coderivatives of single-valued mappings and sum rules for the directionally subdifferential of Lipschitz continuous functions, we first derive necessary conditions for local minimizers in terms of the directionally Mordukhovich subdifferentials to a mathematical programming problem with the set-valued objective functions. We then use the obtained results to derive necessary optimality conditions for a multiobjective discrete-time optimal control problem via the directionally generalized differentiation.