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An efficient iterative method for solving split variational inclusion problem with applications

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  • A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.

    Mathematics Subject Classification: Primary: 47H05, 47J20, 47J25; Secondary: 65K15.

    Citation:

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  • Figure 1.  Comparative results for random instances.

    Figure 2.  Results of the compared algorithms with different cases of initial points.

    Figure 3.  Original test images of Monarch, Flowers and Colorchecker.

    Figure 4.  Original cropped test images of Monarch, Flowers and Colorchecker.

    Figure 5.  Degraded and restored Monarch images by the compared algorithms.

    Figure 6.  Degraded and restored cropped Monarch images by the compared algorithms.

    Figure 7.  Degraded and restored Flowers images by the compared algorithms.

    Figure 8.  Degraded and restored cropped Flowers images by the compared algorithms.

    Figure 9.  Degraded and restored Colorchecker images by the compared algorithms.

    Figure 10.  Degraded and restored cropped Colorchecker images by the compared algorithms.

    Table 1.  The PNSR and SSIM values of the compared algorithms

    Scheme 51 Algorithm 4.4
    Images SNR SSIM SNR SSIM
    Monarch 43.3255 0.9684 39.5788 0.9624
    Flowers 40.6001 0.9116 36.7745 0.8660
    Colorchecker 41.7454 0.8996 39.0134 0.9061
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