A hybrid proximal-extragradient algorithm with inertial effects

TL;DR

This paper introduces an inertial-enhanced hybrid proximal-extragradient algorithm for finding zeros of maximally monotone operators, demonstrating its convergence and unifying several classical algorithms within this framework.

Contribution

It develops a new inertial hybrid proximal-extragradient algorithm and proves its convergence, extending existing methods and embedding classical algorithms into this unified scheme.

Findings

The inertial algorithm converges under extended Fejér monotonicity.

Classical algorithms are special cases of the proposed framework.

The convergence analysis uses Opial's Lemma.

Abstract

We incorporate inertial terms in the hybrid proximal-extragradient algorithm and investigate the convergence properties of the resulting iterative scheme designed for finding the zeros of a maximally monotone operator in real Hilbert spaces. The convergence analysis relies on extended Fej\'er monotonicity techniques combined with the celebrated Opial Lemma. We also show that the classical hybrid proximal-extragradient algorithm and the inertial versions of the proximal point, the forward-backward and the forward-backward-forward algorithms can be embedded in the framework of the proposed iterative scheme.