We study the behavior of the minimal sets of a sequence of convex sets {A_n} 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–Painlev´e 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]) , l_p, and L_p (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

MIGLIERINA, ENRICO;MOLHO, ELENA
2005

Abstract

We study the behavior of the minimal sets of a sequence of convex sets {A_n} 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–Painlev´e 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]) , l_p, and L_p (R) for 1 ≤ p≤∞). We also make a comparison with the existing results related to the topics considered in our work.
Stability; vector optimization; minimal points; set-convergences; convex sets
Miglierina, Enrico; Molho, Elena
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1496200
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