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

Gang Cai; Yekini Shehu; Olaniyi S. Iyiola

doi:10.3934/jimo.2021095

Article Contents

2022, Volume 18, Issue 4: 2873-2902. Doi: 10.3934/jimo.2021095

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

1.
School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China
3.
Department of Mathematics and Physical Sciences, , California University of Pennsylvania, PA, USA

^* Corresponding author: G. Cai
^* Corresponding author: G. Cai

Received: September 2020

Revised: January 2021

Early access: May 2021

Published: July 2022

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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Abstract

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.

Keywords:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Figure 25. Example 2: Case I

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Figure 26. Example 2: Case II

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Figure 27. Example 2: Case III

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Figure 28. Example 2: Case IV

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Figure 29. Example 2: Case V

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Figure 30. Example 2: Case VI

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 $

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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

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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

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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

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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 $

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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

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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

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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

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