T-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities

Abstract

In this paper, we first define a T-cone which is a unified version of several cones, namely, contingent cone, radial cone, C-tangent cone, Clarke tangent cone, S-cone, adjacent cone, etc. Then, we define the T-epiderivative of a set-valued map which includes the contingent epiderivative, radial epiderivative, S-epiderivative, adjacent epiderivative etc. as special cases. We present several properties of such an epiderivative. The generalized vector T-variational inequality problem is also considered. We provide necessary and sufficient conditions for a solution of a set optimization problem. Several existence results for solutions of set optimization problems and a generalized vector T-variational inequality problem are given.