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Analysis on Newton projection method for the split feasibility problem

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Abstract

In this paper, based on a merit function of the split feasibility problem (SFP), we present a Newton projection method for solving it and analyze the convergence properties of the method. The merit function is differentiable and convex. But its gradient is a linear composite function of the projection operator, so it is nonsmooth in general. We prove that the sequence of iterates converges globally to a solution of the SFP as long as the regularization parameter matrix in the algorithm is chosen properly. Especially, under some local assumptions which are necessary for the case where the projection operator is nonsmooth, we prove that the sequence of iterates generated by the algorithm superlinearly converges to a regular solution of the SFP. Finally, some numerical results are presented.

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Acknowledgements

This research was partly supported by the National Natural Science Foundation of China (11271226, 10971118, 11271233), and the Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (BS2012SF027). The authors would like to thank two anonymous referees for their helpful suggestions on the earlier version of this paper.

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Authors and Affiliations

  1. Institute of Operations Research, Qufu Normal University, Rizhao, 276826, Shandong, People’s Republic of China

    Biao Qu & Changyu Wang

  2. Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Naihua Xiu

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  1. Biao Qu
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  2. Changyu Wang
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  3. Naihua Xiu
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Correspondence to Biao Qu.

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Qu, B., Wang, C. & Xiu, N. Analysis on Newton projection method for the split feasibility problem. Comput Optim Appl 67, 175–199 (2017). https://doi.org/10.1007/s10589-016-9884-3

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