An inexact inertial projective splitting algorithm with strong convergence

TL;DR

This paper introduces a novel inexact inertial projective splitting algorithm with strong convergence guarantees for solving composite monotone inclusion problems, including variants with forward-backward steps, and provides iteration-complexity analysis.

Contribution

It presents a new strongly convergent inexact inertial PS algorithm that handles structured problems and includes iteration-complexity results without inertial effects.

Findings

Algorithm achieves strong convergence of iterates.

Iteration-complexity results are established without inertial terms.

Variants with forward-backward steps are developed for structured problems.

Abstract

We propose and study a strongly convergent inexact inertial projective splitting (PS) algorithm for finding zeros of composite monotone inclusion problems involving the sum of finitely many maximal monotone operators. Strong convergence of the iterates is ensured by projections onto the intersection of appropriately defined half-spaces, even in the absence of inertial effects. We also establish iteration-complexity results for the proposed PS method, which likewise hold without requiring inertial terms. The algorithm includes two inertial sequences, controlled by parameters satisfying mild conditions, while preserving strong convergence and enabling iteration-complexity analysis. Furthermore, for more structured monotone inclusion problems, we derive two variants of the main algorithm that employ forward-backward and forward-backward-forward steps.