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Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces

  • *Corresponding author: Yuanheng Wang

    *Corresponding author: Yuanheng Wang 

This work was supported by the NSF of China (12171435, 11971216)

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  • In this paper, we propose a new inertial viscosity iterative algorithm for solving the variational inequality problem with a pseudo-monotone operator and the fixed point problem involving a nonexpansive mapping in real Hilbert spaces. The advantage of the proposed algorithm is that it can work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence of the sequence generated by the proposed algorithm is proved under some suitable assumptions imposed on the parameters. Some numerical experiments are given to support our main results.

    Mathematics Subject Classification: Primary: 47H09, 47J20; Secondary: 65K15.

    Citation:

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  • Figure 1.  The behavior of our Algorithm 3.1 in Example 4.1 ($ m = 200 $)

    Figure 2.  The behavior of our Algorithm 3.1 in Example 4.2 ($ x_{0} = x_{1} = 10\exp(t) $)

    Figure 3.  The behavior of our Algorithm 3.1 in Example 4.3 ($ m = 500000 $)

    Table 1.  Numerical results of all algorithms with different dimensions in Example 4.1

    Algorithms $ m=20 $ $ m=50 $ $ m=100 $ $ m=200 $
    $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
    Our Alg. 3.1 4.47E-05 0.0303 1.70E-04 0.0316 3.17E-04 0.0707 5.00E-04 0.1119
    CTC Alg. 3.1 3.36E-04 0.0227 9.95E-04 0.0227 1.69E-03 0.0250 2.64E-03 0.0273
    GTT Alg. 3.1 2.80E-03 0.0310 6.86E-03 0.0361 1.08E-02 0.1044 1.63E-02 0.1393
    GTT Alg. 3.2 3.38E-03 0.0483 9.78E-03 0.0379 1.64E-02 0.0704 2.56E-02 0.1027
     | Show Table
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    Table 2.  Numerical results of all algorithms with different initial values in Example 4.2

    Algorithms $ x_{1}=10t^{3} $ $ x_{1}=10\sin(2t) $ $ x_{1}=10\log(t) $ $ x_{1}=10\exp(t) $
    $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
    Our Alg. 3.1 1.47E-15 37.7392 4.04E-15 38.5818 7.26E-15 39.9932 3.16E-15 44.6711
    CTC Alg. 3.1 3.16E-13 22.8652 2.55E-13 24.3823 3.36E-12 25.0381 4.39E-12 30.0801
    GTT Alg. 3.1 9.04E-06 33.7345 1.54E-05 34.5784 2.31E-05 36.9545 2.01E-05 45.1488
    GTT Alg. 3.2 6.25E-11 33.5898 7.40E-10 36.0191 1.04E-09 37.5059 1.06E-09 44.7099
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical results of all algorithms with different dimensions in Example 4.3

    Algorithms $ m=500 $ $ m=5000 $ $ m=50000 $ $ m=500000 $
    $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU $ D_{n} $ CPU
    Our Alg. 3.1 7.13E-57 0.0249 8.76E-57 0.1079 3.77E-57 0.4058 8.89E-57 13.8430
    CDP Alg. 3.1 3.97E-27 0.0406 7.89E-27 0.1274 7.25E-27 0.5290 4.78E-26 13.9558
    TSI Alg. 3 8.38E-13 0.0318 7.96E-13 0.1270 8.17E-13 0.4180 6.62E-13 15.0426
    RTDLD Alg. 4 4.72E-10 0.0312 3.07E-07 0.1132 1.64E-03 0.4540 2.59E-02 19.2562
     | Show Table
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