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Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

  • * Corresponding author: G. Cai

    * Corresponding author: G. Cai 

The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504) and the Program of Chongqing Innovation Research Group Project in University (Grant no. CXQT19018)

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  • In this paper, we propose a new inertial Tseng's extragradient iterative algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz operator in real Hilbert spaces. We prove that the sequence generated by proposed algorithm converges strongly to an element of solutions of variational inequality problem under some suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results. The main results obtained in this paper extend and improve some related works in the literature.

    Mathematics Subject Classification: Primary: 47H09, 47H10; Secondary: 47J20, 65K15.

    Citation:

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  • Figure 1.  Example 1: $ k = 20 $, $ N = 10 $

    Figure 2.  Example 1: $ k = 20 $, $ N = 20 $

    Figure 3.  Example 1: $ k = 20 $, $ N = 30 $

    Figure 4.  Example 1: $ k = 20 $, $ N = 40 $

    Figure 5.  Example 1: $ k = 30 $, $ N = 10 $

    Figure 6.  Example 1: $ k = 30 $, $ N = 20 $

    Figure 7.  Example 1: $ k = 30 $, $ N = 30 $

    Figure 8.  Example 1: $ k = 30 $, $ N = 40 $

    Figure 9.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 10) $

    Figure 10.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 21) $

    Figure 11.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 30) $

    Figure 12.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 40) $

    Figure 13.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 10) $

    Figure 14.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 20) $

    Figure 15.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 30) $

    Figure 16.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 40) $

    Figure 17.  Example 1: Different $ \mu $ with $ (N, k) = (20, 10) $

    Figure 18.  Example 1: Different $ \mu $ with $ (N, k) = (20, 20) $

    Figure 19.  Example 1: Different $ \mu $ with $ (N, k) = (20, 30) $

    Figure 20.  Example 1: Different $ \mu $ with $ (N, k) = (20, 40) $

    Figure 21.  Example 1: Different $ \mu $ with $ (N, k) = (30, 10) $

    Figure 22.  Example 1: Different $ \mu $ with $ (N, k) = (30, 20) $

    Figure 23.  Example 1: Different $ \mu $ with $ (N, k) = (30, 30) $

    Figure 24.  Example 1: Different $ \mu $ with $ (N, k) = (30, 40) $

    Figure 25.  Example 2: Case I

    Figure 26.  Example 2: Case II

    Figure 27.  Example 2: Case III

    Figure 28.  Example 2: Case IV

    Figure 29.  Example 2: Case V

    Figure 30.  Example 2: Case VI

    Figure 31.  Example 2: Case I with different $ \gamma $

    Figure 32.  Example 2: Case II with different $ \gamma $

    Figure 33.  Example 2: Case III with different $ \gamma $

    Figure 34.  Example 2: Case IV with different $ \gamma $

    Figure 35.  Example 2: Case V with different $ \gamma $

    Figure 36.  Example 2: Case VI with different $ \gamma $

    Figure 37.  Example 2: Case I with different $ \mu $

    Figure 38.  Example 2: Case II with different $ \mu $

    Figure 39.  Example 2: Case III with different $ \mu $

    Figure 40.  Example 2: Case IV with different $ \mu $

    Figure 41.  Example 2: Case V with different $ \mu $

    Figure 42.  Example 2: Case VI with different $ \mu $

    Figure 43.  The value of error versus the iteration numbers for Example 3

    Table 1.  Methods Parameters Choice for Comparison

    Proposed Alg. $ \epsilon_n = \frac{1}{n^2} $ $ \theta = 0.1 $ $ l=0.001 $ $ \beta_n = \frac{1}{n} $
    $ \gamma = 0.99 $ $ \mu = 0.99 $
    Thong Alg. (1) $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.1 $ $ \beta_n = \frac{1}{n + 1} $ $ \lambda = \frac{1}{1.01L} $
    Thong Alg. (2) $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
    Thong Alg. (3) $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
    Gibali Alg. $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
     | Show Table
    DownLoad: CSV

    Table 2.  Example 1: Comparison among methods with different values of $ N $ and $ k $

    $ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
    $ k=20 $ Iter. Time Iter. Time Iter. Time Iter. Time
    Proposed Alg. 3 1.3843 3 1.7672 3 1.7564 4 2.2017
    Thong Alg. (1) 76 1.2902 139 2.7111 111 2.1715 232 37.7743
    Thong Alg. (2) 2136 36.6812 1561 30.7776 1370 31.8672 1160 4.0453
    Thong Alg. (3) 86 1.1655 152 2.3615 148 2.4878 178 29.0789
    Gibali Alg. 150 12.0085 235 20.3243 319 41.0421 315 4.2520
    $ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
    $ k=30 $ Iter. Time Iter. Time Iter. Time Iter. Time
    Proposed Alg. 3 1.3819 3 1.7834 3 1.6555 3 1.7517
    Thong Alg. (1) 72 1.1548 142 2.6436 136 2.888 207 4.4416
    Thong Alg. (2) 1771 30.2921 1325 28.4023 1132 28.6053 920 26.5714
    Thong Alg. (3) 101 1.5058 90 1.4923 156 2.9515 162 3.6149
    Gibali Alg. 203 17.1568 255 30.849 282 31.6244 303 35.2953
     | Show Table
    DownLoad: CSV

    Table 3.  Example 1 Comparison: Proposed Alg. with different values $ \gamma $

    $ (N, k) $ $ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
    $ (20, 10) $ No. of Iterations 3 3 3 3
    CPU (Time) 1.6337 1.4830 1.4773 1.3843
    $ (20, 20) $ No. of Iterations 3 3 4 3
    CPU (Time) 1.4606 1.4876 2.3980 1.7672
    $ (20, 30) $ No. of Iterations 4 5 4 3
    CPU (Time) 1.8664 0.85257 1.7597 1.7564
    $ (20, 40) $ No. of Iterations 4 3 4 4
    CPU (Time) 1.6573 1.5935 1.8008 2.2017
    $ (30, 10) $ No. of Iterations 3 3 3 3
    CPU (Time) 1.3060 1.3376 1.4359 1.3819
    $ (30, 20) $ No. of Iterations 3 4 3 3
    CPU (Time) 1.4630 1.7306 1.5115 1.7834
    $ (30, 30) $ No. of Iterations 4 3 4 3
    CPU (Time) 1.7102 1.6399 1.7931 1.6555
    $ (30, 40) $ No. of Iterations 5 3 4 3
    CPU (Time) 2.7099 1.6589 2.3287 1.7517
     | Show Table
    DownLoad: CSV

    Table 4.  Example 1 Comparison: Proposed Alg. with different values $ \mu $

    $ (N, k) $ $ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
    $ (20, 10) $ No. of Iterations 3 3 4 3
    CPU (Time) 1.4409 1.4632 2.6888 1.3843
    $ (20, 20) $ No. of Iterations 3 3 3 3
    CPU (Time) 1.5248 1.4840 1.5217 1.7672
    $ (20, 30) $ No. of Iterations 4 3 5 3
    CPU (Time) 1.9571 1.5852 1.9322 1.7564
    $ (20, 40) $ No. of Iterations 6 4 5 4
    CPU (Time) 3.0365 1.8605 2.0718 2.2017
    $ (30, 10) $ No. of Iterations 3 3 3 3
    CPU (Time) 1.3524 1.3416 1.3648 1.3819
    $ (30, 20) $ No. of Iterations 3 3 4 3
    CPU (Time) 1.5265 1.5336 1.6929 1.7834
    $ (30, 30) $ No. of Iterations 5 5 3 3
    CPU (Time) 2.9525 2.0816 1.6424 1.6555
    $ (30, 40) $ No. of Iterations 3 4 8 3
    CPU (Time) 1.6958 2.0833 4.5199 1.7517
     | Show Table
    DownLoad: CSV

    Table 5.  Methods Parameters Choice for Comparison

    Proposed Alg. $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.5 $ $ l=0.01 $ $ \beta_n = \frac{1}{n + 1} $ $ \gamma = 0.99 $ $ \mu = 0.99 $
    Gibali Alg. $ \alpha_n = \frac{1}{1 + n} $ $ l=0.01 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
     | Show Table
    DownLoad: CSV

    Table 6.  Example 2: Prop. Alg. vs Gibali Alg. (Unaccel. Alg.)

    No. of Iterations CPU Time
    Prop. Alg. Gibali Alg. Prop. Alg. Gibali Alg.
    Case I 17 1712 0.001243 0.1244
    Case II 17 1708 0.001518 0.1248
    Case III 17 1713 0.001261 0.1276
    Case IV 17 1729 0.001202 0.1297
    Case V 17 1715 0.001272 0.1258
    Case VI 18 1835 0.001339 0.1564
     | Show Table
    DownLoad: CSV

    Table 7.  Example 2 Comparison: Proposed Alg. with different values $ \mu $

    $ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
    Case I No. of Iterations 17 17 17 17
    CPU (Time) 0.0011992 0.0012179 0.0013264 0.0012430
    Case II No. of Iterations 17 17 17 17
    CPU (Time) 0.0011457 0.0011586 0.0015604 0.0015181
    Case III No. of Iterations 17 17 17 17
    CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606
    Case IV No. of Iterations 17 17 17 17
    CPU (Time) 0.0010843 0.0010928 0.0011176 0.0012022
    Case V No. of Iterations 17 17 17 17
    CPU (Time) 0.0012491 0.0011169 0.0012293 0.0012719
    Case VI No. of Iterations 18 18 18 18
    CPU (Time) 0.0012431 0.0013496 0.0011613 0.0013392
     | Show Table
    DownLoad: CSV

    Table 8.  Example 2 Comparison: Proposed Alg. with different values $ \gamma $

    $ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
    Case I No. of Iterations 17 17 17 17
    CPU (Time) 0.0013518 0.0012097 0.0011754 0.0012430
    Case II No. of Iterations 17 17 17 17
    CPU (Time) 0.0012701 0.0011233 0.0012382 0.0015181
    Case III No. of Iterations 17 17 17 17
    CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606
    Case IV No. of Iterations 17 17 17 17
    CPU (Time) 0.0011530 0.0013917 0.0015395 0.0012022
    Case V No. of Iterations 17 17 17 17
    CPU (Time) 0.0011413 0.0011319 0.0011286 0.0012719
    Case VI No. of Iterations 17 17 18 18
    CPU (Time) 0.0011094 0.0011839 0.0013550 0.0013392
     | Show Table
    DownLoad: CSV
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