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New iterative regularization methods for solving split variational inclusion problems

  • * Corresponding author: Dang Van Hieu (dangvanhieu@tdtu.edu.vn)

    * Corresponding author: Dang Van Hieu (dangvanhieu@tdtu.edu.vn) 
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  • The paper proposes some new iterative algorithms for solving a split variational inclusion problem involving maximally monotone multi-valued operators in a Hilbert space. The algorithms are constructed around the resolvent of operator and the regularization technique to get the strong convergence. Some stepsize rules are incorporated to allow the algorithms to work easily. An application of the proposed algorithms to split feasibility problems is also studied. The computational performance of the new algorithms in comparison with others is shown by some numerical experiments.

    Mathematics Subject Classification: Primary: 65Y05, 65K15; Secondary: 90C33.


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  • Figure 1.  Example 1 for $ M = 100, N = 200 $. Number of iteration is 1393, 1395, 1365, 1414, 1402, respectively

    Figure 2.  Example 1 for M = 200, N = 500. Number of iteration is 202, 204, 198, 209, 210, respectively

    Figure 3.  Example 2 for M = 1024, N = 256. Number of iteration is 763, 759, 228, 748, 756, respectively

    Figure 4.  Example 2 for M = 2048, N = 512. Number of iteration is 357,358,171,359,353, respectively

    Figure 5.  Example 3 for x0(t) = 1. Number of iteration is 350,361,189,414,340, respectively

    Figure 6.  Example 3 for x0(t) = exp(−t). Number of iteration is 238,217,119,335,283, respectively

    Algorithm 1:
    Initialization: Take $ x_0 \in \mathcal{H}_1 $ and two sequences $ \left\{\lambda_n\right\},\,\left\{\alpha_n\right\} \subset \left(0,+\infty\right) $.
    Iterative Steps:
        Step 1. Compute $ x_{n+1}=J_{\lambda_n\mathcal{B}_1}\left(x_n-\lambda_n\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\lambda_n\alpha_n\mathcal{F}x_n\right). $
        Step 2. Set $ n:=n+1 $ and return to Step 1.
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    Algorithm 2:
    Initialization: Take $ x_0 \in \mathcal{H}_1 $ and $ \lambda_0>0 $, $ \mu \in (0,1) $. Choose a sequence $ \left\{\alpha_n\right\} \subset \left(0,+\infty\right) $ such that conditions $ \rm (C2) - (C4) $ hold.
    Iterative Steps:
        Step 1. Compute
    $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\{ \begin{array}{ll} y_n = J_{\lambda_n\mathcal{B}_1}\left(x_n-\lambda_n\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\lambda_n\alpha_n\mathcal{F}x_n\right).\\ x_{n+1} = y_n+\lambda_n\left(\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n\right). \end{array} \right. $
        Step 2. Update $ \lambda_n $:
    $ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \lambda_{n+1} = \min \left\{\lambda_n,\frac{\mu ||x_n-y_n||}{||\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n||}\right\}.$
     | Show Table
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