Genericity of well-posed vector optimization problems

Abstract

In this paper we consider well-posedness properties of vector optimization problems with objective function $f: X \to Y$ where $X$ and $Y$ are Banach spaces and $Y$ is partially ordered by a closed convex pointed cone with nonempty interior. The vector well-posedness notion considered in this paper is the one due to Dentcheva and Helbig, which is a natural extension of Tykhonov well-posedness for scalar optimization problems. When a scalar optimization problem is considered it is possible to prove that under some assumptions the set of functions for which the related optimzation problem is well-posed is dense or even more in "big" i.e. contains a dense $G_{\delta}$ set (these results are called genericity results). The aim of this paper is to extend these genericity results to vector optimization problems.