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Linear Regularity and Linear Convergence of Projection-Based Methods for Solving Convex Feasibility Problems

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Abstract

For a finite/infinite family of closed convex sets with nonempty intersection in Hilbert space, we consider the (bounded) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem. Several sufficient conditions are provided to ensure the bounded linear regularity in terms of the interior-point conditions and some finite codimension assumptions. A unified projection method, called Algorithm B-EMOPP, for solving the convex feasibility problem is proposed, and by using the bounded linear regularity, the linear convergence results for this method are established under a new control strategy introduced here.

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Acknowledgements

The authors are grateful to the anonymous referee for his valuable suggestions and remarks, especially for providing the relevant references [8, 9, 26], which helped to improve the presentation of the paper. The Xiaopeng Zhao was supported in part by the National Natural Science Foundation of China (Grant 11626168). The Kung Fu Ng was supported in part by an Earmarked Grant (GRF) from the Research Grant Council of Hong Kong (Project Nos. CUHK 14304014 and 14302516), and CUHK direct Grant 4053218. The Chong Li was supported in part by the National Natural Science Foundation of China (Grant 11571308). The Jen-Chih Yao was in part supported by the National Science Council of Taiwan under Grant MOST 105-2115-M-039-002-MY3.

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Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, People’s Republic of China

    Xiaopeng Zhao

  2. Department of Mathematics, Chinese University of Hong Kong, Hong Kong, People’s Republic of China

    Kung Fu Ng

  3. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Chong Li

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

    Jen-Chih Yao

  5. Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 807, Taiwan

    Jen-Chih Yao

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  1. Xiaopeng Zhao
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  2. Kung Fu Ng
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  3. Chong Li
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  4. Jen-Chih Yao
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Correspondence to Chong Li.

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Zhao, X., Ng, K.F., Li, C. et al. Linear Regularity and Linear Convergence of Projection-Based Methods for Solving Convex Feasibility Problems. Appl Math Optim 78, 613–641 (2018). https://doi.org/10.1007/s00245-017-9417-1

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