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A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem

  • * Corresponding author: Oluwatosin Temitope Mewomo

    * Corresponding author: Oluwatosin Temitope Mewomo

The first author acknowledge with thanks the bursary and financial support from African Institute for Mathematical Sciences (AIMS), South Africa. The second author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF CoE-MaSS) Doctoral Bursary. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903)

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  • In this paper, we introduce and study a modified extragradient algorithm for approximating solutions of a certain class of split pseudo-monotone variational inequality problem in real Hilbert spaces. Using our proposed algorithm, we established a strong convergent result for approximating solutions of the aforementioned problem. Our strong convergent result is obtained without prior knowledge of the Lipschitz constant of the pseudo-monotone operator used in this paper, and with minimized number of projections per iteration compared to other results on split variational inequality problem in the literature. Furthermore, numerical examples are given to show the performance and advantage of our method as well as comparing it with related methods in the literature.

    Mathematics Subject Classification: 47H09; 47H10; 49J20; 49J40.

    Citation:

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  • Figure 1.  The behavior of $ \mbox{TOL}_n $ with $ \varepsilon = 10^{-3} $ for Example 4.1: Top Left: Case 1; Top Right: Case 2; Bottom Left: Case 3; Bottom Right: Case 4

    Figure 2.  The behavior of $ \mbox{TOL}_n $ with $ \varepsilon = 10^{-5} $ for Example 4.2: Top Left: Case Ⅰ; Top Right: Case Ⅱ; Bottom Left: Case Ⅲ; Bottom Right: Case Ⅳ

    Table 1.  Numerical results for Example 4.1 with $ \varepsilon = 10^{-3} $

    Cases Algorithm 3.1 Algorithm (41) Algorithm (11) Algorithm (42)
    1 CPU
    Iter.
    0.8353
    10
    1.3347
    15
    1.7686
    20
    6.0884
    37
    2 CPU
    Iter.
    0.8529
    8
    1.0366
    13
    1.1735
    17
    3.7446
    30
    3 CPU
    Iter.
    7.3932
    11
    10.0706
    16
    11.5588
    20
    117.5588
    38
    4 CPU
    Iter.
    9.1057
    15
    13.6644
    19
    14.2521
    23
    181.3317
    45
     | Show Table
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    Table 2.  Numerical results for Example 4.2 with $\varepsilon = 10^{-5}$

    Cases Algorithm 3.1 Algorithm (41) Algorithm (11) Algorithm (42)
    1 CPU
    Iter.
    0.1683
    15
    1.0294
    23
    1.0699
    36
    1.1343
    71
    2 CPU
    Iter.
    0.0124
    12
    0.1060
    17
    1.0093
    34
    1.0293
    66
    3 CPU
    Iter.
    0.0146
    9
    0.1125
    10
    1.0122
    29
    1.1248
    55
    4 CPU
    Iter.
    0.0124
    14
    1.0118
    22
    1.1103
    35
    1.2245
    69
     | Show Table
    DownLoad: CSV
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