$E$-subdifferential of $E$-convex Functions and its Applications to Minimization Problem

In this paper, a new concept of the subdifferential is defined for nondifferentiable (not necessarily) locally Lipschitz functions. Namely, the concept of $E$-subdifferential and the notion of $E$-subconvexity are introduced for $E$-convex functions. Thus, the notion of an $E$-subdifferentiable $E$-convex function is introduced and some properties of this class of nondifferentiable nonconvex functions are studied. The necessary optimality conditions in $E$-subdifferentials terms of the involved functions are established for a new class of nondifferentiable optimization problems. The introduced concept of $E$-subconvexity is used to prove the sufficiency of the aforesaid necessary optimality conditions for nondifferentiable optimization problems in which the involved functions are $E$-subdifferentiable $E$-convex.