Clarke generalized directional derivative involving Minkowski difference with applications to set optimization problems

This paper focuses on optimality conditions for an approximate weak minimal solution of a set optimization problem by using a new kind of Clarke generalized directional derivative. First, by means of a Gerstewitz scalarizing function for a partial set order, we introduce a new notion of generalized directional derivative for set-valued maps, discuss its relationships with an existing derivative, and obtain some of its properties, such as Lipschitz continuity. Then, based on the introduced derivative, we establish the optimality conditions for an approximate weak minimal solution. Some of our results improve the corresponding ones in recent literature.