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Finitely convergent iterative methods with overrelaxations revisited

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Abstract

We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation parameters converge to zero, but with a rate slower than any geometric sequence. Unlike other works in this area, which require divergent series of overrelaxations, our approach allows us to consider some summable series. By employing quasi-Fejérian analysis in the latter case, we obtain additional asymptotic convergence guarantees, even when the interior of the solution set is empty.

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Acknowledgements

We are very grateful to an anonymous referee for pertinent comments and helpful suggestions.

Funding

This research was supported by the Israel Science Foundation (Grants Nos. 389/12 and 820/17), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.

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Authors and Affiliations

  1. Department of Computer Science, The Technion—Israel Institute of Technology, 32000, Haifa, Israel

    Victor I. Kolobov

  2. Department of Mathematics, The Technion—Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich & Rafal Zalas

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  1. Victor I. Kolobov
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  2. Simeon Reich
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  3. Rafal Zalas
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Correspondence to Rafal Zalas.

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Kolobov, V.I., Reich, S. & Zalas, R. Finitely convergent iterative methods with overrelaxations revisited. J. Fixed Point Theory Appl. 23, 57 (2021). https://doi.org/10.1007/s11784-021-00888-8

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