<p>This paper focuses on second-order optimality conditions for a weak minimal solution of a set optimization problem by using a new kind of generalized second-order radial derivatives. Firstly, we introduce a new notion of generalized second-order outer radial derivative for set-valued maps by utilizing Minkowski difference, discuss its relationships to some existed derivatives, and obtain some of its properties, such as sum and chain rules. Then, based on the introduced derivative, we establish the optimality conditions for a weak minimal solution. Finally, we apply the results to the problems of uncertain multi-objective programming and shortest path. Some of our results improve and imply the corresponding ones in the recent literature.</p>