A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

TL;DR

This paper introduces a novel primal-dual splitting algorithm for composite monotone inclusions that minimizes the problem's dimensionality without extra operator evaluations, with proven convergence and practical applications.

Contribution

It presents the first minimal lifting primal-dual splitting algorithm for composite monotone inclusions, extending the minimal lifting theorem to schemes with resolvent parameters.

Findings

Algorithm reduces problem dimension compared to existing methods

Convergence of the proposed scheme is rigorously proven

Effective in image deblurring and denoising tasks

Abstract

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.