Construction of Best Bregman Approximations in Reflexive Banach Spaces

An iterative method is proposed to construct the Bregman projection of a point onto a countable intersection of closed convex sets in a reflexive Banach space. 1. Problem statement Let (X , ‖ · ‖) be a reflexive real Banach space with dual (X ∗, ‖ · ‖∗) and let f : X → ]−∞,+∞] be a lower semicontinuous (l.s.c.) convex function which is Gâteaux differentiable on int dom f 6= Ø and Legendre [1, Def. 5.2], i.e., it satisfies the following two properties: (i) ∂f is both locally bounded and single-valued on its domain (essential smoothness); (ii) (∂f)−1 is locally bounded on its domain and f is strictly convex on every convex subset of dom ∂f (essential strict convexity). The Bregman distance associated with f is (1.1) D : X × X → [0,+∞] (x, y) 7→ { f(x)− f(y)− 〈x− y,∇f(y)〉 , if y ∈ int dom f ; +∞, otherwise. Let x0 be a point in X and (Si)i∈I a countable family of closed and convex subsets of X such that (1.2) x0 ∈ int dom f, (int dom f) ∩ ⋂ i∈I Si 6= Ø, and S = dom f ∩ ⋂