On the convergence rate improvement of a splitting method for finding the resolvent of the sum of maximal monotone operators

TL;DR

This paper introduces an accelerated, strongly convergent splitting method for finding the resolvent of the sum of maximal monotone operators in Hilbert spaces, achieving an improved convergence rate of O(1/k).

Contribution

It develops a new, implementable splitting algorithm employing acceleration techniques, enhancing convergence speed for solving operator resolvent problems.

Findings

Convergence rate of O(1/k) demonstrated.

Method is strongly convergent.

Applicable to certain optimization problems.

Abstract

This paper provides a new way of developing the splitting method which is used to solve the problem of finding the resolvent of the sum of maximal monotone operators in Hilbert spaces. By employing accelerated techniques developed by Davis and Yin (in Set-Valued Var. Anal. 25(4):829-858, 2017), this paper presents an implementable, strongly convergent splitting method which is designed to solve the problem. In particular, we show that the distance between the sequence of iterates and the solution converges to zero at a rate O(1/k) to illustrate the efficiency of the proposed method, where k is the number of iterations. Then, we apply the result to a class of optimization problems.