Gerstewitz nonlinear scalar functional and the applications in vector optimization problems

Ying Gao, Liping Tang

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.

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