Convergence Analysis of an Inexact Three-Operator Splitting Algorithm

Chunxiang Zong,Yuchao Tang,Yeol Cho

doi:10.3390/sym10110563

Abstract

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature.

Highlights

Operator splitting algorithms have been widely applied for solving various convex optimization problems in signal and image processing, medical image reconstruction, machine learning and others
It is worth mentioning that Raguet et al [20] introduced a generalized forward-backward splitting (GFBS) algorithm for solving monotone inclusion of the sum of a finite family of maximally monotone operators and a cocoercive operator
In the context of convex minimization problem, Combettes and Wajs [26] introduced an inexact forward-backward splitting algorithm. They analyzed the convergence of the algorithm, which was based on the fixed point theoretic framework in [36]

Summary

Introduction

Operator splitting algorithms have been widely applied for solving various convex optimization problems in signal and image processing, medical image reconstruction, machine learning and others. It is worth mentioning that Raguet et al [20] introduced a generalized forward-backward splitting (GFBS) algorithm for solving monotone inclusion of the sum of a finite family of maximally monotone operators and a cocoercive operator. In the context of convex minimization problem, Combettes and Wajs [26] introduced an inexact forward-backward splitting algorithm They analyzed the convergence of the algorithm, which was based on the fixed point theoretic framework in [36]. An accelerated inexact forward-backward splitting algorithm was proposed by Villa et al [39], who proved that the objective function values have a convergence rate 1/k2 when the allowable error is a certain type, and presented a global analysis of iteration-complexity.

Preliminaries

An Inexact Three-Operator Splitting Algorithm

Conclusions