Radial Epiderivatives Research Articles

Radial epiderivatives and set-valued optimization

In this article we study some important properties of the radial epiderivatives for single-valued and set-valued maps. The relationships between this kind of a derivative and weak subdifferentials and directional derivatives in the single-valued non-convex case has been established. For optimization problems with a single-valued and a set-valued objective function, necessary and sufficient optimality conditions based on the concept of the radial epiderivatives are proved without convexity conditions.

On Weak Subdifferentials, Directional Derivatives, and Radial Epiderivatives for Nonconvex Functions

In this paper we study relations between the directional derivatives, the weak subdifferentials, and the radial epiderivatives for nonconvex real-valued functions. We generalize the well-known theorem that represents the directional derivative of a convex function as a pointwise maximum of its subgradients for the nonconvex case. Using the notion of the weak subgradient, we establish conditions that guarantee equality of the directional derivative to the pointwise supremum of weak subgradients of a nonconvex real-valued function. A similar representation is also established for the radial epiderivative of a nonconvex function. Finally the equality between the directional derivatives and the radial epiderivatives for a nonconvex function is proved. An analogue of the well-known theorem on necessary and sufficient conditions for optimality is drawn without any convexity assumptions.

Radial Epiderivatives and Asymptotic Functions in Nonconvex Vector Optimization

The notions of lower and upper radial epiderivatives (a radial notion different from that generally discussed) for nonconvex vector-valued functions are introduced, many of their properties are established, and some of the optimality conditions for a point to be an ideal, Pareto, or weak-Pareto minimum involving these epiderivatives are also presented. In particular, a characterization for the ideal minima in terms of the lower radial epiderivative is proved. Such a result appears to be new in the literature. Under some mild assumptions on the given function, it is proved that the asymptotic cone of its epigraph is the epigraph of its upper radial epiderivative. Moreover, given a vector minimization problem, we describe the asymptotic behavior of its solution set by introducing some cones of asymptotic directions of the function involved. Finally, we define the lower and upper radial subdifferentials and express the optimality conditions by means of these subdifferentials. Certainly these optimality conditions subsume various necessary or sufficient conditions found in the literature for convex or nonconvex functions.

Optimality conditions in non-convex set-valued optimization

The notion of radial epiderivative is introduced and then a necessary and sufficient condition for a point to be a weak minimal solution (weak-efficient solution) for a non-convex set-valued optimization problem is derived. Such a condition subsumes various necessary and/or sufficient conditions found in the literature for single-valued convex/non-convex mappings.