Higher order properly efficient points in vector optimization

Abstract

We consider the constrained vector optimization problem minC f(x), g(x) - K, where f: ℝn → ℝm and g : ℝn → ℝp are given functions and C ℝm and K ℝp are closed convex cones. Two type of solutions are important for our considerations, namely i-minimizers (isolated minimizers) of order k and pminimizers (properly efficient points) of order k (see e.g. [11]). Every i-minimizer of order k ≥ 1 is a p-minimizer of order k. For k = 1, conditions under which the reversal of this statement holds have been given in [11]. In this paper we investigate the possible reversal of the implication i-minimizer → p-minimizer in the case k = 2. To carry on this study, we develop second-order optimality conditions for p-minimizers, expressed by means of Dini derivatives. Together with the optimality conditions obtained in [11] and [12] in the case of i-minimizers, they play a crucial role in the investigation. Further, to get a satisfactory answer to the posed reversal problem, we deal with sense I and sense II solution concepts, as defined in [11] and [5]. © 2006 Springer-Verlag Berlin Heidelberg.