Stability results for convergence of convex sets and functions in nonreflexive spaces

Abstract

Let Γ(X) be the convex proper lower semicontinuous functions on a normed linear space X. We show, subject to Rockafellar's constraints qualifications, that the operations of sum, episum and restriction are continuous with respect to the slice topology that reduces to the topology of Mosco convergence for reflexive X. We show also when X is complete that the epigraphical difference is continuous. These results are applied to convergence of convex sets.

Characterizations of convex approximate subdifferential calculus in Banach spaces

“…This approach guarantees that the sum f k + g k slice (or Mosco when the underlying space is reflexive) converges whenever both f k and g k converge in the same sense. These results also require conditions on the continuity of the infconvolution of the conjugates together with some constraint qualifications in the spirit of Moreau-Rockafellar's condition (see, e.g., [5], [13], [22], [23], [26], [31]). Consequently, using results on quantitative stability of the subdifferentials (see, e.g., [1], [8], [29]), it can be deduced that the subdifferential of f k +g k (but not necessarily the sum of subdifferentials) converges in the sense of Painlevé-Kuratowski to the subdifferential of f + g. It is important to observe that the approach in this paper leads to characterizations of the approximate subdifferential of f + g by means of those of each approximating function.…”

“…√ δ so that the first two required properties of the lemma follow. Now, by replacing x by x 0 in (23) and using the definition of A we get (by taking x = x δ in (22))…”