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Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems

  • * Corresponding author: Lateef Olakunle Jolaoso

    * Corresponding author: Lateef Olakunle Jolaoso 
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  • Using the concept of Bregman divergence, we propose a new subgradient extragradient method for approximating a common solution of pseudo-monotone and Lipschitz continuous variational inequalities and fixed point problem in real Hilbert spaces. The algorithm uses a new self-adjustment rule for selecting the stepsize in each iteration and also, we prove a strong convergence result for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. Finally, we provide some numerical examples to illustrate the performance and accuracy of our algorithm in finite and infinite dimensional spaces.

    Mathematics Subject Classification: Primary:65K15, 90C33;Secondary:47J25.


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  • Figure 1.  Example 1, Top Left: Case I; Top Right: Case II, Bottom Left: Case III, Bottom Right: Case IV

    Figure 2.  Example 2, Top Left: $ m = 5 $; Top Right: $ m = 15 $, Bottom: $ m = 30 $

    Table 1.  Computation result for Example 1

    Algorithm 4 Algorithm 1 Algorithm 3
    Case I Iter. 5 12 29
    Time 0.6406 1.0043 0.7661
    Case II Iter. 12 45 49
    Time 3.0910 9.5282 3.3343
    Case III Iter. 10 22 39
    Time 1.1391 3.0101 1.7377
    Case IV Iter. 13 56 53
    Time 0.8596 3.9885 1.8918
     | Show Table
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    Table 2.  Computation result for Example 2

    Algorithm 4 Algorithm 3
    $ m=5 $ Iter. 7 11
    Time 0.0036 0.0050
    $ m=15 $ Iter. 8 13
    Time 0.0052 0.0099
    $ m=30 $ Iter. 8 27
    Time 0.0255 0.0884
     | Show Table
    DownLoad: CSV
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