Two New Generalized Radial Epiderivatives in Set-Valued Analysis

The generalized radial epiderivative for nonconvex set-valued maps is formulated in terms of minimal points and therefore does not separately reflect weak and Benson proper minimality. To extend this framework to these two efficiency concepts, we introduce the weak generalized radial epiderivative and the proper generalized radial epiderivative by using weakly minimal and Benson properly minimal points, respectively. We establish existence results for generalized radial epiderivatives under regularity and C-lower-boundedness assumptions and investigate the relationships among the generalized, weak generalized, proper generalized, and radial epiderivatives. In particular, conditions under which the generalized radial epiderivative coincides with the radial epiderivative are obtained. We also provide scalarization-based characterizations of the weak and proper generalized radial epiderivatives by means of augmented dual cones and strongly monotonically increasing functionals. The examples illustrate that the proposed weak and proper notions may differ from the generalized radial epiderivative, showing that they provide distinct extensions of the existing framework.