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The Combination Projection Method for Solving Convex Feasibility Problems

Author

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  • Songnian He

    (Tianjin Key Laboratory for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin 300300, China)

  • Qiao-Li Dong

    (Tianjin Key Laboratory for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin 300300, China)

Abstract

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some x * ∈ C : = ∩ i = 1 m { x ∈ H | c i ( x ) ≤ 0 } , where m is a positive integer, H is a real Hilbert space, and { c i } i = 1 m are convex functions defined as H . The key of the CPM is that, for the current iterate x k , the CPM firstly constructs a new level set H k through a convex combination of some of { c i } i = 1 m in an appropriate way, and then updates the new iterate x k + 1 only by using the projection P H k . We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods.

Suggested Citation

  • Songnian He & Qiao-Li Dong, 2018. "The Combination Projection Method for Solving Convex Feasibility Problems," Mathematics, MDPI, vol. 6(11), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:11:p:249-:d:182252
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    References listed on IDEAS

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    1. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    2. Hong-Kun Xu, 2011. "Averaged Mappings and the Gradient-Projection Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 360-378, August.
    3. Aviv Gibali & Karl-Heinz Küfer & Daniel Reem & Philipp Süss, 2018. "A generalized projection-based scheme for solving convex constrained optimization problems," Computational Optimization and Applications, Springer, vol. 70(3), pages 737-762, July.
    4. Songnian He & Caiping Yang, 2013. "Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, May.
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    Cited by:

    1. Bing Tan & Shanshan Xu & Songxiao Li, 2020. "Modified Inertial Hybrid and Shrinking Projection Algorithms for Solving Fixed Point Problems," Mathematics, MDPI, vol. 8(2), pages 1-12, February.

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