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doi: 10.3934/naco.2021011
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A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

* Corresponding author: Oluwatosin Temitope Mewomo

Received  November 2020 Revised  March 2021 Early access March 2021

Fund Project: 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)

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.

Citation: Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021011
References:
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H. A. AbassC. IzuchukwuF. U. Ogbuisi and O. T. Mewomo, An iterative method for solution of finite families of split minimization problems and fixed point problems, Novi Sad J. Math., 49 (2019), 117-136.   Google Scholar

[2]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2020), 545-574.  doi: 10.1080/02331934.2020.1723586.  Google Scholar

[3]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, A general iterative method for finding common fixed point of finite family of demicontractive mappings with accretive variational inequality problems in Banach spaces, Nonlinear Stud., 27 (2020), 1-24.  doi: 10.1007/s40314-019-1014-2.  Google Scholar

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T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces, Afr. Mat., 2021. doi: 10.1007/s13370-020-00869-z.  Google Scholar

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T. O. Alakoya, L.O. Jolaoso, A. Taiwo and O. T. Mewomo, Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, 2021. doi: 10.1080/02331934.2021.1895154.  Google Scholar

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T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 2021. doi: 10.1007/s11565-020-00354-2.  Google Scholar

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R. I. BotE. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60.  doi: 10.1016/j.ejor.2020.04.035.  Google Scholar

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Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

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C. E. Chidume and M. O. Nnakwe, Iterative algorithms for split variational inequalities and generalized split feasibility problems with applications, J. Nonlinear Var. Anal., 3 (2019), 127-140.   Google Scholar

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H. DehghanC. IzuchukwuO. T. MewomoD. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., 43 (2020), 975-998.  doi: 10.2989/16073606.2019.1593255.  Google Scholar

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G. Fichéra, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 34 (1963), 138-142.   Google Scholar

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A. Gibali, L. O. Jolaoso, O. T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math., 75 (2020), Art. No. 179, 36 pp. doi: 10.1007/s00025-020-01306-0.  Google Scholar

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E. C. Godwin, C. Izuchukwu and O.T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 2020. doi: 10.1007/s40574-020-00.  Google Scholar

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B-S. HeZ-H. Yang and X-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374.  doi: 10.1016/j.jmaa.2004.04.068.  Google Scholar

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D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96.  doi: 10.1007/s10589-016-9857-6.  Google Scholar

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C. IzuchukwuK. O. AremuA. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.  Google Scholar

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C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), Art. No. 98, 23 pp. doi: 10.1007/s11784-020-00834-0.  Google Scholar

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C. IzuchukwuC. C. Okeke and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed point problems for countable families of multivalued type-one demicontractive-type mappings, Ukran. Mat. Zh., 71 (2019), 1480-1501.   Google Scholar

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C. IzuchukwuG.C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

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L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), Art. No. 38, 28 pp. doi: 10.1007/s40314-019-1014-2.  Google Scholar

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L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766.   Google Scholar

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P. E. Maing$\acute{e}$ and M. L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168.  doi: 10.1007/s10957-016-0972-4.  Google Scholar

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G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 2021. doi: 10.1007/s11075-021-01081-1.  Google Scholar

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G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, On $\theta$-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math., 53 (2020), 95-111.  doi: 10.1515/dema-2020-0006.  Google Scholar

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show all references

References:
[1]

H. A. AbassC. IzuchukwuF. U. Ogbuisi and O. T. Mewomo, An iterative method for solution of finite families of split minimization problems and fixed point problems, Novi Sad J. Math., 49 (2019), 117-136.   Google Scholar

[2]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2020), 545-574.  doi: 10.1080/02331934.2020.1723586.  Google Scholar

[3]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, A general iterative method for finding common fixed point of finite family of demicontractive mappings with accretive variational inequality problems in Banach spaces, Nonlinear Stud., 27 (2020), 1-24.  doi: 10.1007/s40314-019-1014-2.  Google Scholar

[4]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces, Afr. Mat., 2021. doi: 10.1007/s13370-020-00869-z.  Google Scholar

[5]

T. O. Alakoya, L.O. Jolaoso, A. Taiwo and O. T. Mewomo, Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, 2021. doi: 10.1080/02331934.2021.1895154.  Google Scholar

[6]

T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 2021. doi: 10.1007/s11565-020-00354-2.  Google Scholar

[7]

R. I. BotE. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60.  doi: 10.1016/j.ejor.2020.04.035.  Google Scholar

[8]

F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.  doi: 10.1016/0022-247X(67)90085-6.  Google Scholar

[9]

C. Byrne, A unified treatment for some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar

[10]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.  Google Scholar

[11]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.   Google Scholar

[12]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[13]

Y. CensorT. ElfvingN. Kopf and T. Bortfield, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[14]

L. C. CengN. Hadjisavvas and N-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646.  doi: 10.1007/s10898-009-9454-7.  Google Scholar

[15]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[16]

C. E. Chidume and M. O. Nnakwe, Iterative algorithms for split variational inequalities and generalized split feasibility problems with applications, J. Nonlinear Var. Anal., 3 (2019), 127-140.   Google Scholar

[17]

H. DehghanC. IzuchukwuO. T. MewomoD. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., 43 (2020), 975-998.  doi: 10.2989/16073606.2019.1593255.  Google Scholar

[18]

G. Fichéra, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 34 (1963), 138-142.   Google Scholar

[19]

A. Gibali, L. O. Jolaoso, O. T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math., 75 (2020), Art. No. 179, 36 pp. doi: 10.1007/s00025-020-01306-0.  Google Scholar

[20]

E. C. Godwin, C. Izuchukwu and O.T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 2020. doi: 10.1007/s40574-020-00.  Google Scholar

[21]

B-S. HeZ-H. Yang and X-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374.  doi: 10.1016/j.jmaa.2004.04.068.  Google Scholar

[22]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96.  doi: 10.1007/s10589-016-9857-6.  Google Scholar

[23]

C. IzuchukwuK. O. AremuA. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.  Google Scholar

[24]

C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), Art. No. 98, 23 pp. doi: 10.1007/s11784-020-00834-0.  Google Scholar

[25]

C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 2020. doi: 10.1080/02331934.2020.1808648.  Google Scholar

[26]

C. IzuchukwuC. C. Okeke and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed point problems for countable families of multivalued type-one demicontractive-type mappings, Ukran. Mat. Zh., 71 (2019), 1480-1501.   Google Scholar

[27]

C. IzuchukwuG.C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

[28]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), Art. No. 38, 28 pp. doi: 10.1007/s40314-019-1014-2.  Google Scholar

[29]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766.   Google Scholar

[30]

P. D. Khanh and P. T. Vuong, Modified projection method for strongly pseudo-monotone variational inequalities, J. Global Optim., 58 (2014), 341-350.   Google Scholar

[31]

G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekon. Mat. Metody, 12 (1976), 747-756.   Google Scholar

[32]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2.  Google Scholar

[33]

P. E. Maing$\acute{e}$ and M. L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168.  doi: 10.1007/s10957-016-0972-4.  Google Scholar

[34]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Anal., 72 (2010), 2086-2099.  doi: 10.1016/j.na.2009.10.009.  Google Scholar

[35]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27 (2020), 127-152.  doi: 10.36045/bbms/1590199308.  Google Scholar

[36]

G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 2021. doi: 10.1007/s11075-021-01081-1.  Google Scholar

[37]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, On $\theta$-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math., 53 (2020), 95-111.  doi: 10.1515/dema-2020-0006.  Google Scholar

[38]

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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
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
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
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
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