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

Lateef Olakunle Jolaoso; Maggie Aphane

doi:10.3934/jimo.2020178

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2022, Volume 18, Issue 2: 773-794. 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

Lateef Olakunle Jolaoso^, and
Maggie Aphane

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

^* Corresponding author: Lateef Olakunle Jolaoso
^* Corresponding author: Lateef Olakunle Jolaoso

Received: March 2020

Revised: September 2020

Early access: December 2020

Published: March 2022

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Abstract

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.

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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

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

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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

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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

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