Primal-dual splitting for structured composite monotone inclusions with or without cocoercivity

TL;DR

This paper introduces a versatile primal-dual splitting algorithm for complex structured monotone inclusions, improving convergence and flexibility over existing methods, with applications demonstrated in cancer detection.

Contribution

It unifies and extends existing algorithms for structured monotone inclusions, reducing dimensionality and allowing larger stepsizes with a single convergence analysis.

Findings

The algorithm accommodates various cocoercivity and Lipschitz constants.

It achieves a larger stepsize range than recent methods.

Numerical experiments validate its effectiveness in cancer detection.

Abstract

In this paper, we propose a primal-dual splitting algorithm for a broad class of structured composite monotone inclusions that involve finitely many set-valued operators, compositions of set-valued operators with bounded linear operators, and single-valued operators possibly without cocoercivity. The proposed algorithm is not only a unification for several contemporary algorithms but also a blueprint to generate new algorithms with graph-based structures using a single transparent convergence analysis. Our approach reduces dimensionality compared with the standard product space technique, which typically reformulates the original problem as the sum of two maximally monotone operators in order to apply splitting methods. It accommodates different cocoercive or Lipschitz constants as well as different resolvent parameters, and yields a larger allowable stepsize range than recent methods. We demonstrate the practicality of the approach by a numerical experiment on cancer detection using the decentralized fused LASSO problem.