In this paper, we study the properties of Gerstewitz nonlinear scalar
functional with respect to coradiant 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. 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 approximate solutions with respect to
co-radiant set of vector optimization problem. We also state that this
special functional can be used as a coherent measure in the portfolio
problem.