On the rate of convergence of iterated Bregman projections and of the alternating algorithm

Abstract

We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces. For Banach spaces which are convex and smooth of power type, we exhibit a condition which implies linear convergence of this method. We show these convergence results for iterates of Bregman projections onto closed linear subspaces. Using an intimate connection between the metric projection onto a closed linear subspace and the Bregman projection onto its annihilator, we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated Bregman projection method.

Convergence Results for Contractive Type Set-Valued Mappings

“…In T : X → 2 X \ {∅}, T(x) is closed for each x ∈ X, and X 0 ⊂ X is a nonempty compact set such that T(X 0 ) ⊂ X 0 (2) and where, for every x ∈ X 0 and each y ∈ X, H(T(x), T(y)) ≤ ϕ(ρ(x, y))ρ(x, y).…”

Fixed Point and Convergence Results for Contractive-Type Self-Mappings of Metric Spaces with Graphs

“…For more than sixty years, the fixed point theory of nonlinear operators has been an important area of nonlinear analysis. One of its main topics is the study of the existence of fixed points of nonexpansive and contractive maps [1][2][3][4][5][6][7][8][9][10][11][12]. It should be mentioned that the analysis of nonexpansive operators acting on complete metric spaces with graphs is of great interest [13][14][15][16][17][18][19][20][21].…”

Three Existence Results in the Fixed Point Theory

“…The work referenced here [1] was the starting point of the fixed point theory of nonexpansive maps, which is a growing field of research; see [2][3][4][5][6][7]. In particular, the convergence of iterated Bregman projections and of the alternating algorithm were studied in [2], fixed point theorems under Mizoguchi-Takahashi-type conditions were obtained in [3], Ciric-type results were proved in [4], fixed point theory in modular function spaces was discussed in [5] and cyclic contractions were analyzed in [6]. Note that the research on nonexpansive maps in spaces with graphs is of great importance; see [8][9][10][11][12] and the references mentioned therein.…”