Abstract
The notion of Fejér 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 Fejér 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.
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Acknowledgements
We thank Terry Rockafellar for his inspirational talk at the virtual West Coast Optimization Meeting in May 2021 which stimulated this research. We also thank Walaa Moursi for suggesting to investigate the operator T in Sect. 5. HHB and XW were partially supported by NSERC Discovery Grants. MKL was partially supported by NSERC Discovery grants of HHB and XW and SERB-UBC fellowship.
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Bauschke, H.H., Krishan Lal, M. & Wang, X. Directional asymptotics of Fejér monotone sequences. Optim Lett 17, 531–544 (2023). https://doi.org/10.1007/s11590-022-01896-4
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DOI: https://doi.org/10.1007/s11590-022-01896-4