American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021185
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New iterative regularization methods for solving split variational inclusion problems

 1 TIMAS - Thang Long University, Ha Noi, Vietnam 2 Institute of Mathematics, VAST, Hanoi, 18 Hoang Quoc Viet, Hanoi, Vietnam 3 Department of Basic Sciences, College of Air Force, Nha Trang City, Vietnam

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

Received  September 2020 Revised  March 2021 Early access November 2021

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.

Citation: Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021185
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References:
Example 1 for $M = 100, N = 200$. Number of iteration is 1393, 1395, 1365, 1414, 1402, respectively
Example 1 for M = 200, N = 500. Number of iteration is 202, 204, 198, 209, 210, respectively
Example 2 for M = 1024, N = 256. Number of iteration is 763, 759, 228, 748, 756, respectively
Example 2 for M = 2048, N = 512. Number of iteration is 357,358,171,359,353, respectively
Example 3 for x0(t) = 1. Number of iteration is 350,361,189,414,340, respectively
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.
 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.
 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 $\lambda_n$: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \lambda_{n+1} = \min \left\{\lambda_n,\frac{\mu ||x_n-y_n||}{||\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n||}\right\}.$
 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 $\lambda_n$: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \lambda_{n+1} = \min \left\{\lambda_n,\frac{\mu ||x_n-y_n||}{||\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n||}\right\}.$
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