Convergence Results of Forward-Backward Algorithms for Sum of Monotone Operators in Banach Spaces

TL;DR

This paper extends forward-backward algorithms to Banach spaces for finding zeros of sums of maximal monotone and Lipschitz continuous monotone operators, providing convergence results that encompass many existing problems.

Contribution

It introduces convergence analysis of splitting methods for these operators in Banach spaces, generalizing previous Hilbert space results.

Findings

Established weak and strong convergence results

Unified several existing problem cases as special instances

Extended applicability of forward-backward algorithms

Abstract

It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper.