Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces

By Dan Butnariu, Elena Resmerita. 2006

304 citations recorded by the source corpus.

The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9df6869da189407c280ea5d293cb3ee33aca14cd

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In Computational Theory and Mathematics

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In Numerical Analysis

Citations in the corpus

Nonlinear iterative methods for linear ill-posed problems in Banach spaces (2006, methodology): “Recently, some nonlinear iterative methods have been proposed in the context of Bregman projections and the minimization of convex functionals, and they are shown to be weakly convergent [ 5 , 15].”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “Following the literature (see [141]) we say a function f is uniformly convex at the point x0 if δf (x0, ε) > 0 for each ε > 0.”
Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces (2010, background): “Recall that the function f is called sequentially consistent (see [17]) if for any two sequences xn n∈ and yn n∈ in X such that the first one is bounded,”
A Characterization of Bregman Firmly Nonexpansive Operators Using a New Monotonicity Concept (2011, background): “See, for instance, [2, 3, 4, 5, 6, 13, 12, 7, 14, 15, 17, 20, 21, 22, 23, 24] and the references therein.”
Right Bregman Nonexpansive Operators in Banach Spaces (2012, methodology): “Examples of totally convex functions can be found, for instance, in [11, 18, 19].”
Inexact Proximal Point Methods for Equilibrium Problems in Banach Spaces (2007, background): “H2 is known to hold when g is lower semicontinuous and uniformly convex on bounded sets (see [10]).”
Uniformly convex functions on Banach spaces (2008, background): “On the other hand, [4] shows that if ‖ · ‖ is uniformly convex, then the function f(x) = ‖x‖ for r > 1 is totally convex which is weaker than f being uniformly convex; see [3, 5, 6] for applications of totally convex and other related convex functions.”