On radially epidifferentiable functions and regularity conditions in nonconvex optimization

Abstract: In this paper we study the radial epiderivative notion for nonconvex functions, which extends the (classical) directional derivative concept. The paper presents new definition and new properties for this notion and establishes relationships between the radial epiderivative, the Clarke's directional derivative, the Rockafellar's subderivative and the directional derivative. The radial epiderivative notion is used to establish new regularity conditions without convexity conditions. The paper analyzes necessary and sufficient conditions for global optimums in nonconvex optimization via the generalized derivatives studied in this paper. We establish a necessary and sufficient condition for a descent direction for radially epidifferentiable nonconvex functions. The paper presents explicit formulations for computing the weak subgradients in terms of the radial epiderivatives and vice versa, which are very important from point of view of developing solution methods for finding global optimums in nonsmooth and nonconvex optimization. All the properties and theorems presented in this paper, are illustrated and interpreted on examples.