A new subdifferential for dealing with nonconvex functions is provided in the following paper and the usual properties are presented as well. Furthermore, characterizations and optimality conditions for a point to be a solution for the nonconvex minimization problem are given. In particular, new KKT-type optimality conditions for nonconvex nonsmooth constraint optimization problems are developed. Moreover, a relationship with the proximity operator for lower semicontinuous quasiconvex functions is given and, as a consequence, the nonemptiness of this subdifferential for large classes of quasiconvex functions is ensured.