We study the behavior of the minimal sets of a sequence of convex sets {An} converging to a given set A. The main feature of the present work is the use of convexity properties of the sets An 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 ([a, b]) , lp, and Lp (R) for 1 ≤ p≤∞). We also make a comparison with the existing results related to the topics considered in our work.
Convergence of minimal sets in convex vector optimization
MOLHO, ELENA;
2005-01-01
Abstract
We study the behavior of the minimal sets of a sequence of convex sets {An} converging to a given set A. The main feature of the present work is the use of convexity properties of the sets An 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 ([a, b]) , lp, and Lp (R) for 1 ≤ p≤∞). We also make a comparison with the existing results related to the topics considered in our work.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
