Quantitative results on Fejer monotone sequences

TL;DR

This paper develops explicit quantitative convergence rates for various iterative algorithms satisfying Fejer monotonicity, applicable in metric spaces including CAT(0)-spaces, enhancing understanding of their convergence behavior.

Contribution

It introduces a unified approach to derive explicit metastability rates for Fejer monotone sequences across multiple iterative methods in general metric spaces.

Findings

Provides explicit convergence rates for proximal point algorithms.

Extends results to fixed point iterations in W-hyperbolic and CAT(0) spaces.

Demonstrates applicability to a broad class of iterative procedures.

Abstract

We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejer monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called W-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.