∊-Variational inequalities in partially ordered spaces

Abstract

The variational principle of Ekeland is an integral part of Numerical Analysis. It can be used to deduce necessary conditions for solutions of real-valued optimization problems (cf. [I]. [14]) In this paper we present necessary conditions for approximately efficient elements of vector optimization problems for which the objective function takes its values in a linear topological space. We Investigate a characterization of approximately efficient elements by using the directional derivative and the Gateaux derivation. Moreover, in the convex case we apply the subdifferential calculus in order to derive necessary conditions for approximately efficient elements. These results can be considered as vector-valued ∊-variational inequalities for approximately efficient elements. Moreover, we will give equivalent statements by using a scalarization. Finally, we will illustrate our results on the example of a vecto-valued approxlmatlon problem. for which we derive ∊-Kolmogorov-conditions