Stability of the geometric Ekeland variational principle: Convex case

Abstract

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x 0,f(x 0)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK ? = {(x, ?) ?X × ?:? ?x? ≤ ? ?}, where ? is a fixed positive scalar. In this case, we write (x 0,f(x 0))??-extf. Here, we investigate the following question: if (x 0,f(x 0))??-extf, wheref is a convex function, and if ?f n ? is a sequence of convex functions convergent tof in some sense, can (x 0,f(x 0)) be recovered as a limit of a sequence of points taken from ?-extf n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.