Directional derivative of set-valued mappings involving generalized oriented distance functions and applications to set optimization

This paper focuses on the directional derivative and subdifferential of set-valued mappings via nonlinear scalarizing functions.Firstly, we define the directional derivative and subdifferential of set-valued mappings by using a generalized oriented distance function.Secondly, we systematically investigated the operational rules, positive homogeneity, chain rule, and upper semicontinuity of the directional derivative for set-valued mappings.Thirdly, we examine the convexity and weak * closedness of the subdifferential of set-valued mappings, as well as its relationship with the directional derivative.Finally, the optimality conditions for set optimization problems are established by utilizing the introduced subdifferential.