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An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in hilbert spaces

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  • In this paper, we present extension of a class of split variational inequality problem and fixed point problem due to Lohawech et al. (J. Ineq Appl. 358, 2018) to a class of multiple sets split variational inequality problem and common fixed point problem (CMSSVICFP) in Hilbert spaces. Using the Halpern subgradient extragradient theorem of variational inequality problems, we propose a parallel Halpern subgradient extragradient CQ-method with adaptive step-size for solving the CMSSVICFP. We show that a sequence generated by the proposed algorithm converges strongly to the solution of the CMSSVICFP. We give a numerical example and perform some preliminary numerical tests to illustrate the numerical efficiency of our method.

     

    Note: The name of the fourth author has been corrected from Widaya Kumam to Wiyada Kumam.

    Mathematics Subject Classification: Primary: 47H05, 47H09, 49M37, 65K10.

    Citation:

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  • Figure 1.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 5) $ and TOL = $ 10^{-5} $

    Figure 2.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 5) $ and TOL = $ 10^{-5} $

    Figure 3.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 10) $ and TOL = $ 10^{-5} $

    Figure 4.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 10) $ and TOL = $ 10^{-5} $

    Figure 5.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (10; 10) $ and TOL = $ 10^{-5} $

    Figure 6.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (10; 10) $ and TOL = $ 10^{-5} $

    Figure 7.  Numerical illustration of Algorithm 3.1 with different values of $ \kappa_{n} = \frac{1}{k(n+2)} $

    Figure 8.  Numerical illustration of Algorithm 3.1 with different values of $ \kappa_{n} = \frac{1}{k(n+2)} $

    Figure 9.  Numerical illustration of Algorithm 3.1 with different values of $ \rho_{n} = k $

    Figure 10.  Numerical illustration of Algorithm 3.1 with different values of $ \rho_{n} = k $

    Figure 11.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (1, 3) $ and TOL = $ 10^{-4} $

    Figure 12.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (1, 3) $ and TOL = $ 10^{-4} $

    Figure 13.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (3, 4) $ and TOL = $ 10^{-4} $

    Figure 14.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (3, 4) $ and TOL = $ 10^{-4} $

    Table 1.  Numerical data for Experiment 1

    N.P S(N; M) TOL. CPU(s) ITER.
    $ 1 $ (5; 5) $ 10^{-5} $ 4.66201370000000 68
    $ 2 $ (5; 10) $ 10^{-5} $ 8.59728030000000 76
    $ 3 $ (10; 10) $ 10^{-5} $ 9.57778720000000 92
    $ 4 $ (10; 20) $ 10^{-5} $ 13.4638439200000 108
    $ 5 $ (20; 30) $ 10^{-5} $ 20.5758686930000 123
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical data for Experiment 2

    k S(N; M) TOL. CPU(s) ITER.
    $ 1 $ (5; 10) $ 10^{-5} $ 6.29621610000000 80
    $ 3 $ (5; 10) $ 10^{-5} $ 7.10547680000000 92
    $ 10 $ (5; 10) $ 10^{-5} $ 10.2573273000000 113
    $ 20 $ (5; 10) $ 10^{-5} $ 18.1375856000000 155
    $ 50 $ (5; 10) $ 10^{-5} $ 30.5698815000000 226
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical data for Experiment 3

    k S(N; M) TOL. CPU(s) ITER.
    $ 0.15 $ (5; 10) $ 10^{-5} $ 18.6105856000000 208
    $ 0.65 $ (5; 10) $ 10^{-5} $ 16.1703019000000 167
    $ 1.15 $ (5; 10) $ 10^{-5} $ 11.3304408000000 128
    $ 1.64 $ (5; 10) $ 10^{-5} $ 9.50234320000000 106
    $ 1.89 $ (5; 10) $ 10^{-5} $ 6.33439070000000 67
     | Show Table
    DownLoad: CSV

    Table 4.  Numerical data for Example 4.2

    SAlgorithm Algorithm 3.1
    $ x_{1} $ CPU(s) ITER. CPU(s) ITER.
    $ (1, 3) $ 10.7807425000000 577 3.90895790000000 282
    $ (3, 4) $ 40.4251408000000 1926 8.90772100000000 507
     | Show Table
    DownLoad: CSV
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