ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION

The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big- $O$ rate to little- $o$ without any other rest...

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Bibliographic Details
Published inBulletin of the Australian Mathematical Society Vol. 96; no. 1; pp. 162 - 170
Main Author MATSUSHITA, SHIN-YA
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.08.2017
Subjects
Algorithms
Hilbert space
47H10
Krasnosel’skiĭ–Mann iteration
primary 47H09
47J25
convergence rate
secondary 47H05
proximal point algorithm
nonexpansive
Douglas–Rachford method
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Summary:The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big- $O$ rate to little- $o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.
ISSN:0004-9727
1755-1633
DOI:10.1017/S000497271600109X