Convergence of Minimal Sets in Convex Vector Optimization

Abstract

We study the behavior of the minimal sets of a sequence of convex sets $ \left\{ A_{n}\right\} $ converging to a given set $A.$ The main feature of the present work is the use of convexity properties of the sets $A_{n}$ and $ A$ to obtain upper and lower convergence of the minimal frontiers. We emphasize that we study both Kuratowski--Painlevé convergence and Attouch--Wets convergence of minimal sets. Moreover, we prove stability results that hold in a normed linear space ordered by a general cone, in order to deal with the most common spaces ordered by their natural nonnegative orthants (e.g., $C\left( \left[ a,b\right] \right) ,$ $l^{p}$, and $L^{p}\left( \mathbb{R}\right) $ for $1\leq p\leq \infty $). We also make a comparison with the existing results related to the topics considered in our work.