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 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.