Linear Regularity for a Collection of Subsmooth Sets in Banach Spaces

Abstract

Abstract. Using variational analysis, we study the linear regularity for a collection of finitely many closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting. Moreover the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption.

Unified approach to some geometric results in variational analysis

“…The approximate projection results for a single closed set were first established in [25,27]. Let X ∈ X and let X m denote the product of m copies of X.…”

“…In the Banach space setting, Theorem 3.10(i) is known when ∂ a is the Clarke-Rockafellar subdifferential ∂ c (cf. [27]). To the best of our knowledge, the result given in part (ii) of Theorem 3.10 is new even for a restricted class X of (Banach or Asplund) spaces.…”

BCQ and Strong BCQ for Nonconvex Generalized Equations with Applications to Metric Subregularity

“…For the case that X is an Asplund space, Mordukhovich and Shao [23] have proved that ∂φ(x) = Limsup The following lemmas will be used in our analysis. Readers are invited to consult references [25] and [26] respectively for more details. Lemma 2.1 Let X be a Banach (resp.…”

Metric subregularity for proximal generalized equations in Hilbert spaces

“…In contrast with the approximate projection theorem established in [42] in terms of the Clarke normal cone, we use Lemma 2.3 to establish the following approximate projection result in a Hilbert space in terms of the proximal normal cone.…”