# On inexact relative-error hybrid proximal extragradient, forward-backward and Tseng's modified forward-backward methods with inertial effects

## TL;DR

This paper introduces an inertial under-relaxed relative-error hybrid proximal extragradient method with convergence guarantees, extending to inertial forward-backward and Tseng's methods for structured monotone inclusions, under flexible assumptions.

## Contribution

It develops a novel inertial under-relaxed HPE method with convergence analysis and applies it to advanced forward-backward algorithms for monotone problems.

## Key findings

- Proves asymptotic convergence of the proposed method.
- Establishes nonasymptotic iteration-complexity bounds.
- Demonstrates effectiveness on structured monotone inclusion problems.

## Abstract

In this paper, we propose and study the asymptotic convergence and nonasymptotic global convergence rates (iteration-complexity) of an inertial under-relaxed version of the relative-error hybrid proximal extragradient (HPE) method for solving monotone inclusion problems. We analyze the proposed method under more flexible assumptions than existing ones on the extrapolation and relative-error parameters. As applications, we propose and/or study inertial under-relaxed forward-backward and Tseng's modified forward-backward type methods for solving structured monotone inclusions.