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Applied Mathematics & Optimization

, Volume 78, Issue 3, pp 613–641 | Cite as

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

  • Xiaopeng Zhao
  • Kung Fu Ng
  • Chong LiEmail author
  • Jen-Chih Yao
Article

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.

Keywords

Convex feasibility problem Linear regularity Projection algorithm 

Mathematics Subject Classification

47H09 65J05 41A25 90C25 

Notes

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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Xiaopeng Zhao
    • 1
  • Kung Fu Ng
    • 2
  • Chong Li
    • 3
    Email author
  • Jen-Chih Yao
    • 4
    • 5
  1. 1.Department of MathematicsTianjin Polytechnic UniversityTianjinPeople’s Republic of China
  2. 2.Department of MathematicsChinese University of Hong KongHong KongPeople’s Republic of China
  3. 3.School of Mathematical SciencesZhejiang UniversityHangzhouPeople’s Republic of China
  4. 4.Center for General EducationChina Medical UniversityTaichungTaiwan
  5. 5.Research Center for Nonlinear Analysis and OptimizationKaohsiung Medical UniversityKaohsiungTaiwan

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