Convergence analysis of an inexact inertial Krasnoselskii-Mann algorithm with applications

TL;DR

This paper introduces an inexact inertial Krasnoselskii-Mann algorithm that incorporates errors in updates, providing convergence analysis and applications to monotone inclusion and convex minimization problems.

Contribution

It presents a novel inexact inertial Krasnoselskii-Mann algorithm with convergence guarantees and practical applications, extending existing inertial methods.

Findings

Weak convergence under various conditions

Nonasymptotic convergence rate established

Applications to proximal point and forward-backward splitting algorithms

Abstract

The classical Krasnoselskii-Mann iteration is broadly used for approximating fixed points of nonexpansive operators. To accelerate the convergence of the Krasnoselskii-Mann iteration, the inertial methods were received much attention in recent years. In this paper, we propose an inexact inertial Krasnoselskii-Mann algorithm. In comparison with the original inertial Krasnoselskii-Mann algorithm, our algorithm allows error for updating the iterative sequence, which makes it more flexible and useful in practice. We establish weak convergence results for the proposed algorithm under different conditions on parameters and error terms. Furthermore, we provide a nonasymptotic convergence rate for the proposed algorithm. As applications, we propose and study inexact inertial proximal point algorithm and inexact inertial forward-backward splitting algorithm for solving monotone inclusion problems and the corresponding convex minimization problems.