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Optimal parameter selections for a general Halpern iteration

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Abstract

Let C be a closed affine subset of a real Hilbert space H and \(T:C \rightarrow C\) be a nonexpansive mapping. In this paper, for any fixed u ∈ C, a general Halpern iteration process:

$$\left\{\begin{array}{ll} x_{0} \in C,\\ x_{n + 1}=t_{n}u+(1-t_{n})Tx_{n},n\geq 0, \end{array}\right. $$

is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, \(\mathbb {R}\). The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities.

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Funding

This work was supported by the Fundamental Research Funds for the Central Universities (3122017078).

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Authors and Affiliations

  1. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Songnian He & Tao Wu

  2. Department of Mathematics Education, Gyeongsang National University, Jinju, 660-701, Korea

    Yeol Je Cho

  3. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Yeol Je Cho

  4. Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Themistocles M. Rassias

Authors
  1. Songnian He
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  2. Tao Wu
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  3. Yeol Je Cho
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  4. Themistocles M. Rassias
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Correspondence to Yeol Je Cho.

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He, S., Wu, T., Cho, Y.J. et al. Optimal parameter selections for a general Halpern iteration. Numer Algor 82, 1171–1188 (2019). https://doi.org/10.1007/s11075-018-00650-1

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  • DOI: https://doi.org/10.1007/s11075-018-00650-1

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