Resolvent splitting with minimal lifting for composite monotone inclusions

TL;DR

This paper introduces a new resolvent splitting method with minimal lifting for solving composite monotone inclusions involving multiple operators and linear transformations, optimizing computational efficiency.

Contribution

It proposes a resolvent splitting with minimal lifting for composite monotone inclusions, extending previous work and providing new algorithms for cases with two operators.

Findings

Exact computation of resolvents, linear operators, and adjoints once per iteration

Recovery of previous minimal lifting splitting when the linear operator is identity

New minimal 1-fold lifting splitting for two-operator cases

Abstract

In this paper we propose a resolvent splitting with minimal lifting for finding a zero of the sum of $n\ge 2$ maximally monotone operators involving the composition with a linear bounded operator. The resolvent of each monotone operator, the linear operator, and its adjoint are computed exactly once in the proposed algorithm. In the case when the linear operator is the identity, we recover the resolvent splitting with minimal lifting developed in Malitsky-Tam (2021). We also derive a new resolvent splitting for solving the composite monotone inclusion in the case $n=2$ with minimal $1-$fold lifting.