Extrapolation and local acceleration of an iterative process for common fixed point problems

Abstract

We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point x ∈ H, the hyperplane through T x whose normal is x − T x always "cuts" the space into two half-spaces one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these we conduct convergence analysis of successive iteration algorithms.

Linear convergence rates for extrapolated fixed point algorithms

“…Similar reasoning can be repeated for the extrapolated cyclic cutter with σ k defined in (1.4); see also Corollary 4.3 below. Thus we recover the framework presented in [17].…”

A string averaging method based on strictly quasi-nonexpansive operators with generalized relaxation

“…We have to mention that any strictly relaxed cutter operator is strictly quasi-nonexpansive operator [13,Remark 2.1.44.] but the converse is not true in general, see Example 2.4. We analyze a fixed point iteration method based on generalized relaxation of an sQNE operator which is constructed by averaging of strings and each string is a composition of finitely many sQNE operators. Our analysis indicates that the generalized relaxation of cutter operators is inherently able to provide more acceleration comparing with [15].…”

“…To solve such problems, using iterative projection methods has been suggested by many researchers, see, e.g., [4]. Since the computing of projections is expensive, it is advised to use, easy computing operators such as subgradient projections and T−class operator, which was introduced and investigated in [5] and studied in several research works as [6,15] and references therein. This class, which is named cutter by some authors, see, e.g., [15], contains projection operators, subgradient projections, firmly nonexpansive operators, the resolvents of maximal monotone operators and strongly quasi-nonexpansive operators but it is a subset of strictly quasi-nonexpansive (sQNE) operators, see [15, p. 810] and [6,13] for more details.…”

“…They used a relaxed version of the projection operators in the string averaging procedure whereas we extend it to generalized relaxation of sQNE operators. The generalized relaxation strategy is studied for nonexpansive operators in [12], for cutter operators in [15] and recently for strictly relaxed cutter operators in [49]. We have to mention that any strictly relaxed cutter operator is strictly quasi-nonexpansive operator [13,Remark 2.1.44.] but the converse is not true in general, see Example 2.4. We analyze a fixed point iteration method based on generalized relaxation of an sQNE operator which is constructed by averaging of strings and each string is a composition of finitely many sQNE operators.…”

An acceleration scheme for Dykstra’s algorithm

“…For most alternating projection methods several acceleration schemes have been proposed; see e. g., [2,6,7,12,[14][15][16][17]21,25,28,33,34,37,[40][41][42][43]45,53]. However, as far as we know no effective acceleration scheme has been developed for Dykstra's algorithm.…”