Projective splitting with backward, half-forward and proximal-Newton steps

TL;DR

This paper introduces a novel projective splitting algorithm that efficiently solves complex multi-term composite monotone inclusion problems by combining backward, half-forward, and proximal-Newton steps, with proven weak convergence.

Contribution

It presents a new algorithm that integrates multiple step types for structured monotone problems, expanding the toolkit for solving such inclusions.

Findings

Algorithm achieves weak convergence for complex problems.

Handles multiple operator types within a unified framework.

Demonstrates effectiveness in structured monotone inclusion problems.

Abstract

We propose and study the weak convergence of a projective splitting algorithm for solving multi-term composite monotone inclusion problems involving the finite sum of $n$ maximal monotone operators, each of which having an inner four-block structure: sum of maximal monotone, Lipschitz continuous, cocoercive and smooth differentiable operators. We show how to perform backward and half-forward steps with respect to the maximal monotone and Lipschitz$+$cocoercive components, respectively, while performing proximal-Newton steps with respect to smooth differentiable blocks.