Proper Strict Efficiency in Set Optimization with Partial Set Order Relation

This paper is devoted to the investigation of the proper strict efficient solutions to a set optimization problem with a partial set order relation. Firstly, the notion of proper strict efficient solution defined by the Minkowski difference is introduced, and it is worth mentioning that the introduced strict efficiency is different from those in the existing literature. Secondly, a class of generalized contingent derivatives for set-valued maps is proposed, which are characterized in terms of a set criterion. Finally, the necessary and sufficient optimality conditions and a scalarization theorem for proper strict efficiency are established. Some concrete examples are given to illustrate the obtained results.