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Selective projection methods for solving a class of variational inequalities

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Abstract

Very recently, Gibali et al. (Optimization 66, 417–437 2017) proposed a method, called selective projection method (SPM) in this paper, for solving the variational inequality problem (VIP) defined on \(C:=\bigcap _{i = 1}^{m} C^{i}\neq \emptyset \), where m ≥ 1 is an integer and \(\{C^{i}\}_{i = 1}^{m}\) is a finite level set family of convex functions on a real Hilbert space H. For the current iterate xn, SPM updates xn+ 1 by projecting onto a half-space \(C^{i_{n}}_{n} (\supset C^{i_{n}})\) constructed by using the input data, where in ∈{1,2,⋯ ,m} is selected by a special rule. The prominent advantage of SPM is that it is concise and easy to implement. Gibali et al. proved its convergence in the Euclidean space \(H=\mathbb {R}^{d}\). In this paper, we firstly prove the strong convergence of SPM in a general Hilbert space. The proof given in this paper is very different from that given by Gibali et al. We also extend SPM to solve VIP defined on the common fixed point set of finite nonexpansive self-mappings of H. Then, we estimate the convergence rate of SPM and its extension in the nonasymptotic sense. Finally, we give some preliminary numerical experiments which illustrate the advantage of SPM.

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Acknowledgements

The authors sincerely thank the reviewers for their pertinent comments and good suggestions.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities (3122017072).

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

    Songnian He & Hanlin Tian

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  1. Songnian He
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  2. Hanlin Tian
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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Correspondence to Songnian He.

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He, S., Tian, H. Selective projection methods for solving a class of variational inequalities. Numer Algor 80, 617–634 (2019). https://doi.org/10.1007/s11075-018-0499-x

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  • DOI: https://doi.org/10.1007/s11075-018-0499-x

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