A new representation of a proximal subdifferential by employing a directional derivative

In this paper, a new representation of the proximal subdifferential of a nonsmooth function is presented by using a directional derivative. The upper-semicontinuity property of the proximal subdifferential is proved via the new representation. The existence and necessary conditions of an optimal solution for a class of inf-convolution functions are obtained by using the proximal subdifferential. Finally, a relationship between the proximal subdifferential and the quasidifferential is established. Based on the relation, the proximal subdifferential can be computed easily when the quasidifferential is a polytope.