Gerstewitz nonlinear scalar functional and the applications in vector optimization problems

Abstract

In this paper, we  study the properties of Gerstewitz nonlinear scalar functional with respect to  co-radiant set and radiant set in  real linear space. With the help of nonconvex separation theorem with respect to co-radiant set,  we first obtain that Gerstewitz nonlinear scalar functional is a special co-radiant(radiant) functional when the corresponding set is a co-radiant(radiant) set. And based on the subadditivity property of this functional with respect to the convex co-radiant set, we calculate its Fenchel(approximate) subdifferential. As the applications, we derive the optimality conditions for the $\varepsilon-$efficient points with respect to co-radiant set. And we also state that this special functional can be used as a coherent measure in the portfolio problem.