Directional asymptotics of Fej\'er monotone sequences
Heinz H. Bauschke, Manish Krishan Lal, Xianfu Wang

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsOptimization and Variational Analysis · Fixed Point Theorems Analysis · Mathematical Inequalities and Applications
