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

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