# Convergence of minimal sets in convex vector optimization

Enrico Miglierina, Elena Molho

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)

## 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.
Original language English 513-526 14 SIAM Journal on Optimization 15 https://doi.org/10.1137/030602642 Published - 2005

## Keywords

• Convex sets
• Minimal points
• Set-convergences
• Stability
• Vector Optimization

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