In this paper, we introduce a Hausdorff-type distance relative to an ordering cone between two sets. We obtain some properties of the Hausdorff-type distance. In particular, we give a characterization of the Hausdorff-type distance. Moreover, we introduce the Clarke generalized directional derivative for set-valued mappings by using the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín [Nonconvex scalarization in set optimization with set-valued maps. J Math Anal Appl. 2007;325:1–18]. Some properties of the Clarke generalized directional derivative are given. As applications, we present necessary and sufficient optimality conditions for set optimization problems.

在本文中，我们介绍了相对于两组之间的有序锥面的Hausdorff型距离。我们获得了Hausdorff型距离的一些属性。特别是，我们给出了Hausdorff型距离的特征。此外，我们通过使用l型较少阶关系的非线性标量函数，引入了Clarke广义定向导数，用于set值映射，这是由Hernández和Rodríguez-Marín[Nonconvex scalarization in set Optimization with set-valued map。J数学肛门应用。2007; 325：1-18]。给出了Clarke广义方向导数的一些性质。作为应用程序，我们为集合优化问题提供了必要的充分条件。