A Characterization of Q-Minimal Solutions in Set-Valued Optimization in Terms of Radial Derivatives

Abstract

In this paper we deal with a set-valued optimization problem with an abstract set constraint. We present a new characterization of Q-minimal solutions in terms of higher-order outer and inner radial derivatives introduced by Anh and Khanh in [4]. The Q-minimality includes several kinds of solutions known from the literature (among others, a Borwein-properly efficient point and a Henig-properly efficient point). However, this characterization is difficult to apply in practice, therefore we formulate also other higher-order necessary and sufficient optimality conditions. All the results are stated in terms of higher-order radial derivatives, which are particularly interesting because, contrary to classical derivatives, they lead to global sufficient conditions without any convexity assumptions. At the end, we give an example to show that the presented necessary condition is not sufficient for Q-minimality.