A modular string averaging procedure for solving the common fixed point problem for quasi-nonexpansive mappings in Hilbert space

In this paper we consider the common fixed point problem for a finite family of quasi-nonexpansive mappings U i : ℋ → ℋ , where i ∈ I := {1,…, M }, M ≥1, and ℋ is a real Hilbert space. This problem is defined as follows: find x ∈ ⋂ i ∈ I Fix U i ≠ ∅ , where Fix U i : = { z ∈ ℋ ∣ z = U i z } . We pro...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	Numerical algorithms Ročník 72; číslo 2; s. 297 - 323
Hlavní autori:	Reich, Simeon, Zalas, Rafał
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2016 Springer Nature B.V
Predmet:	Algebra Algorithms Computer Science Convergence Hilbert space Iterative methods Numeric Computing Numerical Analysis Original Paper Strings Theory of Computation Nonexpansive operator Cutter Block iterative algorithms Common fixed point problem Intermittent control Perturbation resilience Cyclic control Superiorization 47H10 Convex feasibility problem 65F10 String averaging Subgradient projection 47H09 65J99 Almost cyclic control Quasi-nonexpansive operator 47J25 47N10 Firmly nonexpansive operator 46N10 46N40
ISSN:	1017-1398, 1572-9265
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Popis
Shrnutí:	In this paper we consider the common fixed point problem for a finite family of quasi-nonexpansive mappings U i : ℋ → ℋ , where i ∈ I := {1,…, M }, M ≥1, and ℋ is a real Hilbert space. This problem is defined as follows: find x ∈ ⋂ i ∈ I Fix U i ≠ ∅ , where Fix U i : = { z ∈ ℋ ∣ z = U i z } . We propose the following iterative method: x 0 ∈ ℋ , x k + 1 : = T k x k , where for each k =0,1,2,…, the operator T k is defined by a certain amalgamation procedure called modular string averaging . The main idea of this procedure is to combine repeatedly and recursively three primal operations: relaxation, convex combination and composition of the given operators U i . The modular string averaging procedure, when combined with the above iterative method, provides a very flexible framework which covers and fills the gap between different algorithmic approaches such as string averaging and block iterative schemes. Moreover, our framework enables us to construct many algorithmic schemes, the convergence of which has not been investigated so far. The aim of this paper is to establish both weak and strong convergence results for the above iterative method. Moreover, in the case of firmly nonexpansive U i ’s, we show that convergence is preserved in the presence of inexact computations. In particular, this implies that the iterative scheme is resilient to bounded perturbations, which is important from the superiorization methodology point of view.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1017-1398 1572-9265
DOI:	10.1007/s11075-015-0045-z