A class of Fejer convergent algorithms, approximate resolvents and the Hybrid Proximal-Extragradient method

TL;DR

This paper introduces a comprehensive framework for Fejer convergent algorithms, unifies several existing methods through approximate resolvents, and demonstrates the convergence of the Hybrid Proximal-Extragradient method within this framework.

Contribution

It develops a general convergence framework for Fejer algorithms, introduces a new concept of approximate resolvents, and unifies multiple splitting methods under this approach.

Findings

Unified view of Forward-Backward, Tseng's, and Korpelevich's methods.

Proved convergence of approximate resolvent-based methods.

Identified the Hybrid Proximal-Extragradient method as an iteration map within this framework.

Abstract

A new framework for analyzing Fejer convergent algorithms is presented. Using this framework we define a very general class of Fejer convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method. We show that methods based on families of approximate resolvents fall within the aforementioned class of Fejer convergent methods. We prove that such approximate resolvents are the iteration maps of the Hybrid Proximal-Extragradient method.