Sectionwise connected sets in vector optimization

We introduce the notion of sectionwise connected set as a new tool to investigate nonconvex vector optimization. Indeed, the image of a K-convex set through a K-quasiconnected vector function is proved to be sectionwise connected. Some properties of the minimal frontiers of sectionwise connected sets are studied in a finite dimensional framework. We prove that in a sectionwise connected set local and globalminimal points coincide. Moreover, every minimal point is also a strict minimal point. This can be considered as a sort of stability property of the minimal frontier of a sectionwise connected set with respect to perturbations of the order structure. Finally, we develop a stability analysis of minimal frontiers of sectionwise connected sets. Indeed, we consider a sequence of sectionwise connected sets converging in the sense of Kuratowski-Painlevé to a given set Q and we prove the lower convergence of the minimal frontiers of the perturbed sets to the minimal frontier of Q.