FAST KRASNOSEL'SKI\u I--MANN ALGORITHM WITH A CONVERGENCE RATE OF THE FIXED POINT ITERATION OF o(1/k)

被引：3

作者：

Bot, Radu Ioan ^{[1
]}

Nguyen, Dang-Khoa ^{[1
]}

机构：

[1] Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

来源：

SIAM JOURNAL ON NUMERICAL ANALYSIS | 2023年 / 61卷 / 06期

基金：

奥地利科学基金会;

关键词：

nonexpansive operator; averaged operator; Krasnosel'ski\u {\i}--Mann iteration; Nesterov's momentum; Lyapunov analysis; convergence rates; convergence of iterates; SPLITTING ALGORITHMS; WEAK-CONVERGENCE; OPTIMIZATION; SEQUENCE;

D O I：

10.1137/22M1504305

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The Krasnosel'skiu --Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skiu --Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of o(1/k) while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.

引用

页码：2813 / 2843

页数：31