A Class of Fejér Convergent Algorithms, Approximate Resolvents and the Hybrid Proximal-Extragradient Method

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 preserves 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, which is a generalization of the classical Proximal Point Algorithm.