On the asymptotic behaviour of the Aragon Artacho-Campoy algorithm

TL;DR

This paper analyzes the asymptotic behavior of the Aragón Artacho-Campoy algorithm, showing convergence to the nearest zero of the sum of two maximally monotone operators and offering new interpretations of the operators involved.

Contribution

It completes the analysis of the algorithm's asymptotic behavior and introduces novel interpretations using resolvent and proximal average concepts.

Findings

The underlying curve converges to the nearest zero of the sum of the operators.

Provides a new interpretation of the operators via resolvent and proximal average.

Extends the analysis to the generalized setting of maximally monotone operators.

Abstract

Arag\'on Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone operators. In this paper, we complete this analysis by demonstrating that the underlying curve converges to the nearest zero of the sum of the two operators. We also provide a new interpretation of the underlying operators in terms of the resolvent and the proximal average.