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The generalized modular string averaging procedure and its applications to iterative methods for solving various nonlinear operator theory problems

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Abstract

A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such methods.

In this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered.

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Funding

Simeon Reich was partially supported by the Israel Science Foundation (Grant 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 Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Kay Barshad & Simeon Reich

  2. Department of Mathematics, Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  3. The Center for Mathematics and Scientific Computation, University of Haifa, 3498838, Haifa, Israel

    Aviv Gibali

Authors
  1. Kay Barshad
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  2. Aviv Gibali
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  3. Simeon Reich
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Contributions

All authors contributed equally with respect to all aspects to this work.

Corresponding author

Correspondence to Aviv Gibali.

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Barshad, K., Gibali, A. & Reich, S. The generalized modular string averaging procedure and its applications to iterative methods for solving various nonlinear operator theory problems. Numer Algor 94, 1797–1818 (2023). https://doi.org/10.1007/s11075-023-01555-4

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  • DOI: https://doi.org/10.1007/s11075-023-01555-4

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