Directional asymptotics of Fej\'er monotone sequences

TL;DR

This paper investigates the directional asymptotic behavior of Fejér monotone sequences, providing new convergence insights and examples illustrating the complexity of their cluster points, especially in infinite-dimensional spaces.

Contribution

It introduces directionally asymptotic results for Fejér monotone sequences and highlights the necessity of weak convergence in infinite-dimensional settings.

Findings

Strongly convergent subsequences have directionally asymptotic properties

Examples show large sets of cluster points can occur

Weak convergence is essential in infinite-dimensional spaces

Abstract

The notion of Fej\'er monotonicity is instrumental in unifying the convergence proofs of many iterative methods, such as the Krasnoselskii-Mann iteration, the proximal point method, the Douglas-Rachford splitting algorithm, and many others. In this paper, we present directionally asymptotical results of strongly convergent subsequences of Fej\'er monotone sequences. We also provide examples to show that the sets of directionally asymptotic cluster points can be large and that weak convergence is needed in infinite-dimensional spaces.