Quantitative results on algorithms for zeros of differences of monotone operators in Hilbert space

TL;DR

This paper uses proof mining techniques to derive quantitative rates of metastability and convergence for algorithms approximating zeros of differences of maximally monotone operators in Hilbert spaces.

Contribution

It introduces new quantitative bounds and generalizes previous results for quasi-Fejér monotone sequences under metric regularity assumptions.

Findings

Provides a rate of metastability for the algorithm.

Establishes a rate of convergence under metric regularity.

Generalizes previous results for Fejér monotone sequences.

Abstract

We provide quantitative information in the form of a rate of metastability in the sense of T. Tao and (under a metric regularity assumption) a rate of convergence for an algorithm approximating zeros of differences of maximally monotone operators due to A. Moudafi by using techniques from `proof mining', a subdiscipline of mathematical logic. For the rate of convergence, we provide an abstract and general result on the construction of rates of convergence for quasi-Fej\'er monotone sequences with metric regularity assumptions, generalizing previous results for Fej\'er monotone sequences due to U. Kohlenbach, G. L\'opez-Acedo and A. Nicolae.