Computing the resolvent of the sum of maximally monotone operators with the averaged alternating modified reflections algorithm

TL;DR

This paper introduces a new strongly convergent splitting algorithm based on the averaged alternating modified reflections method for computing the resolvent of sums of maximally monotone operators, with applications to finite sums.

Contribution

It generalizes the averaged alternating modified reflections algorithm to handle sums of maximally monotone operators and proposes parallel variants for finite sums.

Findings

Proves strong convergence of the new splitting method.

Develops parallel splitting variants for finite sums.

Provides a theoretical foundation for applying the method to complex operator sums.

Abstract

The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it can be used to compute the resolvent of the sum of two maximally monotone operators. This gives rise to a new splitting method, which is proved to be strongly convergent. A standard product space reformulation permits to apply the method for computing the resolvent of a finite sum of maximally monotone operators. Based on this, we propose two variants of such parallel splitting method.