The convergence of level sets and the convergence of E-solutions in terms of p- hausdorff distance

Abstract EN

Let X be a real normed space, and let { /, fn , n e N } be a sequence ; extended real-valued functions on X.

The aim of this paper is to study the ; convergence of the level sets to the sequence (/ ^) at heights ctnin terms of P-Hausdorff distance.

The purpose of this study is to investigate the set of s-optimal solutions to some optimization problems from a metrical point of, view, and generalize some results in literature that deal with a topological point of view.

More specifically, we show that the sequence (/ ^) is p-Hausdorff distance convergent to / if and only if for each s > 0, the sequence I of sets (s-argmin / ^) is p-Hausdorff distance convergent to e-argmin / ).

An analogous result holds for £-suBdifferentials of convex " lower semi-continuous functions defined on a Banch space and also for : £—projctions of a point to! Convex closed subset in X.