# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020178
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## Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems

 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 90 Medunsa 0204, South Africa

* Corresponding author: Lateef Olakunle Jolaoso

Received  March 2020 Revised  September 2020 Early access December 2020

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.

Citation: Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020178
##### References:

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##### References:
Example 1, Top Left: Case I; Top Right: Case II, Bottom Left: Case III, Bottom Right: Case IV
Example 2, Top Left: $m = 5$; Top Right: $m = 15$, Bottom: $m = 30$
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
 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
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
 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

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