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.