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doi: 10.3934/naco.2021037
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Inertial method for split null point problems with pseudomonotone variational inequality problems

1. 

School of Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa

2. 

Mathematics Institute African University of Science and Technology, Abuja Nigeria

3. 

Department of Mathematics, Federal University Dutsinma, Kastina State Nigeria

4. 

Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94 Medunsa 0204, Pretoria, South Africa

*Corresponding author: Christian Chibueze Okeke

Received  April 2021 Revised  September 2021 Early access September 2021

This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.

Citation: Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021037
References:
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H. AttouchJ. Peypouquet and P. Redont, A dynamical approach to an inertial forward backward algorithm for convex minimization, SIAM J. Optim., 24 (2014), 232-256.  doi: 10.1137/130910294.  Google Scholar

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R. I. Bot E. R. Csetnek, An inertial alternating direction method of multipliers, Minimax Theory Appl., 1, (2016) 29-49.  Google Scholar

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C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453. doi: 10.1088/0266-5611/18/2/310.  Google Scholar

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C. ByrneY. CensorA. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

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L.C. CengA. PetruselX. Qin and J. C. Yao, A Modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93-108.  doi: 10.24193/fpt-ro.  Google Scholar

[7]

L. C. CengA. PetruselJ. C. Yao and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501.  doi: 10.24193/fpt-ro.2018.2.39.  Google Scholar

[8]

L. C. CengA. PetruselX. Qin and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337-1358.  doi: 10.1080/02331934.2020.1858832.  Google Scholar

[9]

L. C. Ceng, and M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., 33 (2020), 19 pp. doi: 10.1186/s13660-020-2306-1.  Google Scholar

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L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2021), 715-740.  doi: 10.1080/02331934.2019.1647203.  Google Scholar

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Y. CensorA. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Meth. Softw., 26 (2011), 827-845.  doi: 10.1080/10556788.2010.551536.  Google Scholar

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Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132.  doi: 10.1080/02331934.2010.539689.  Google Scholar

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P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.  Google Scholar

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S. V. DensisovV. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765.  doi: 10.1007/s10559-015-9768-z.  Google Scholar

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M. Eslamian and J. Vahidi, Split common fixed point problem of nonexpansive semigroup, Mediterr J. Math., 13 (2016), 1177-1195.  doi: 10.1007/s00009-015-0541-3.  Google Scholar

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A. Gibali, Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces, J. Nonlinear Var. Anal., 2 (2018), 49-61.   Google Scholar

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A. Gibali, A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces, J. Nonlinear Anal. Optim., 6 (2015), 41-51.   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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A. N. Iusem and B. F. Savaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321.  doi: 10.1080/02331939708844365.  Google Scholar

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P. Jailoka and S. Suantai, Split common fixed point and null point problems for demicontractive operators in Hilbert spaces, Optim. Methods Softw., 34 (2019), 248-263.  doi: 10.1080/10556788.2017.1359265.  Google Scholar

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S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optimization, 68 (2019), 2365-2391.  doi: 10.1080/02331934.2019.1638389.  Google Scholar

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E. V. Khoroshilova, Extragradient-type method for optimal control problem with linear constraints and convex objective function, Optim. Lett., 7 (2013), 1193-1214.  doi: 10.1007/s11590-012-0496-2.  Google Scholar

[26]

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

[27]

P. E. Mainge, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.  doi: 10.1016/j.jmaa.2005.12.066.  Google Scholar

[28]

P. E. Mainge 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

[29]

Y. V. Malitsky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 148 (2014), 318-355.  doi: 10.1007/s10559-014-9614-8.  Google Scholar

[30]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.  Google Scholar

[31]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an Extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.  Google Scholar

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M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7.  Google Scholar

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C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 223-248.   Google Scholar

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C. C. Okeke and O. T. Mewomo, Regularized gradient-projection algorithm for solving one-parameter nonexpansive semigroup, constrained convex minimization and generalized equilibrium problems, Buletinul Academiei Ştiiţe Republicii Moldova Matematica, 88 (2018), 32-56.  Google Scholar

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S. Reich, D. V. Thong, Q. Li Dong, X-H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numer. Algo., 87 (2021), 52-549. doi: 10.1007/s11075-020-00977-8.  Google Scholar

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S. SuantaiN. Pholasa and P. Cholamijiak, The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595-1615.  doi: 10.3934/jimo.2018023.  Google Scholar

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[41]

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D. V. ThongY. Shehu and O. S. Iyiola, A new iterative methods for solving pseudomonotone variational inequality with non-Lipschitz operators, Comp. Appl. Math., 39 (2020), 108.  doi: 10.1007/s40314-020-1136-6.  Google Scholar

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P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.  Google Scholar

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P. T. Vuong, On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities, J. Optim. Theory Appl., 176 (2018), 399-409.  doi: 10.1007/s10957-017-1214-0.  Google Scholar

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S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.  Google Scholar

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Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019). doi: 10.1007/s10092-018-0300-5.  Google Scholar

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Y. Shehu and O. S. Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algo., 76 (2017), 259-282.  doi: 10.1007/s11075-016-0253-1.  Google Scholar

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show all references

References:
[1]

H. AttouchJ. Peypouquet and P. Redont, A dynamical approach to an inertial forward backward algorithm for convex minimization, SIAM J. Optim., 24 (2014), 232-256.  doi: 10.1137/130910294.  Google Scholar

[2]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[3]

R. I. Bot E. R. Csetnek, An inertial alternating direction method of multipliers, Minimax Theory Appl., 1, (2016) 29-49.  Google Scholar

[4]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453. doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[5]

C. ByrneY. CensorA. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[6]

L.C. CengA. PetruselX. Qin and J. C. Yao, A Modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93-108.  doi: 10.24193/fpt-ro.  Google Scholar

[7]

L. C. CengA. PetruselJ. C. Yao and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501.  doi: 10.24193/fpt-ro.2018.2.39.  Google Scholar

[8]

L. C. CengA. PetruselX. Qin and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337-1358.  doi: 10.1080/02331934.2020.1858832.  Google Scholar

[9]

L. C. Ceng, and M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., 33 (2020), 19 pp. doi: 10.1186/s13660-020-2306-1.  Google Scholar

[10]

L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2021), 715-740.  doi: 10.1080/02331934.2019.1647203.  Google Scholar

[11]

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

[12]

R. W. Cottle and J. C. Yao, Pseudomonotone of complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.  Google Scholar

[13]

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

[14]

Y. CensorA. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Meth. Softw., 26 (2011), 827-845.  doi: 10.1080/10556788.2010.551536.  Google Scholar

[15]

Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132.  doi: 10.1080/02331934.2010.539689.  Google Scholar

[16]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.  Google Scholar

[17]

S. V. DensisovV. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765.  doi: 10.1007/s10559-015-9768-z.  Google Scholar

[18]

M. Eslamian and J. Vahidi, Split common fixed point problem of nonexpansive semigroup, Mediterr J. Math., 13 (2016), 1177-1195.  doi: 10.1007/s00009-015-0541-3.  Google Scholar

[19]

A. Gibali, Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces, J. Nonlinear Var. Anal., 2 (2018), 49-61.   Google Scholar

[20]

A. Gibali, A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces, J. Nonlinear Anal. Optim., 6 (2015), 41-51.   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]

A. N. Iusem and B. F. Savaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321.  doi: 10.1080/02331939708844365.  Google Scholar

[23]

P. Jailoka and S. Suantai, Split common fixed point and null point problems for demicontractive operators in Hilbert spaces, Optim. Methods Softw., 34 (2019), 248-263.  doi: 10.1080/10556788.2017.1359265.  Google Scholar

[24]

S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optimization, 68 (2019), 2365-2391.  doi: 10.1080/02331934.2019.1638389.  Google Scholar

[25]

E. V. Khoroshilova, Extragradient-type method for optimal control problem with linear constraints and convex objective function, Optim. Lett., 7 (2013), 1193-1214.  doi: 10.1007/s11590-012-0496-2.  Google Scholar

[26]

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

[27]

P. E. Mainge, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.  doi: 10.1016/j.jmaa.2005.12.066.  Google Scholar

[28]

P. E. Mainge 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

[29]

Y. V. Malitsky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 148 (2014), 318-355.  doi: 10.1007/s10559-014-9614-8.  Google Scholar

[30]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.  Google Scholar

[31]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an Extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.  Google Scholar

[32]

M. S. Nikol'skii, Convergence of the gradient projection method in optimal control problems, Comp. Math. Model., 18 (2007), 148-156.  doi: 10.1007/s10598-007-0015-y.  Google Scholar

[33]

M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7.  Google Scholar

[34]

C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 223-248.   Google Scholar

[35]

C. C. Okeke and O. T. Mewomo, Regularized gradient-projection algorithm for solving one-parameter nonexpansive semigroup, constrained convex minimization and generalized equilibrium problems, Buletinul Academiei Ştiiţe Republicii Moldova Matematica, 88 (2018), 32-56.  Google Scholar

[36]

S. Reich, D. V. Thong, Q. Li Dong, X-H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numer. Algo., 87 (2021), 52-549. doi: 10.1007/s11075-020-00977-8.  Google Scholar

[37]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar

[38]

G. Stampacchia, Forms bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[39]

S. SuantaiN. Pholasa and P. Cholamijiak, The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595-1615.  doi: 10.3934/jimo.2018023.  Google Scholar

[40]

S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), 281-287.  doi: 10.1080/02331934.2015.1020943.  Google Scholar

[41]

W. Takahashi, The split common null point problem in Banach spaces, Arch Math., 104 (2015), 357-365.  doi: 10.1007/s00013-015-0738-5.  Google Scholar

[42]

D. V. ThongY. Shehu and O. S. Iyiola, A new iterative methods for solving pseudomonotone variational inequality with non-Lipschitz operators, Comp. Appl. Math., 39 (2020), 108.  doi: 10.1007/s40314-020-1136-6.  Google Scholar

[43]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.  Google Scholar

[44]

P. T. Vuong, On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities, J. Optim. Theory Appl., 176 (2018), 399-409.  doi: 10.1007/s10957-017-1214-0.  Google Scholar

[45]

P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms, 81 (2019), 26-291.  doi: 10.1007/s11075-018-0547-6.  Google Scholar

[46]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.  Google Scholar

[47]

Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019). doi: 10.1007/s10092-018-0300-5.  Google Scholar

[48]

Y. Shehu and O. S. Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algo., 76 (2017), 259-282.  doi: 10.1007/s11075-016-0253-1.  Google Scholar

[49]

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Figure 1.  Example 5.1, Top Left: Case I; Top Right: Case II; Bottom: Case III
Figure 2.  Example 5.2, Top Left: Case I; Top Right: Case II; Bottom: Case III
Table 1.  Computational result for Example 5.1
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 7 8 8
CPU time (sec) 0.034 0.0050 0.0034
Algo. 3.3 ($ \alpha =3 $) No of Iter. 13 14 14
CPU time (sec) 0.0109 0.0160 0.0076
NUMA No of Iter. 51 54 58
CPU time (sec) 0.0249 0.0296 0.0288
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 7 8 8
CPU time (sec) 0.034 0.0050 0.0034
Algo. 3.3 ($ \alpha =3 $) No of Iter. 13 14 14
CPU time (sec) 0.0109 0.0160 0.0076
NUMA No of Iter. 51 54 58
CPU time (sec) 0.0249 0.0296 0.0288
Table 2.  Computational result for Example 5.2
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 8 16 90
CPU time (sec) 0.0096 0.0223 0.0425
Algo. 3.3 ($ \alpha =3 $) No of Iter. 8 19 323
CPU time (sec) 0.0035 0.00089 0.4394
NUMA No of Iter. 8 17 69
CPU time (sec) 0.0051 0.0084 0.0257
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 8 16 90
CPU time (sec) 0.0096 0.0223 0.0425
Algo. 3.3 ($ \alpha =3 $) No of Iter. 8 19 323
CPU time (sec) 0.0035 0.00089 0.4394
NUMA No of Iter. 8 17 69
CPU time (sec) 0.0051 0.0084 0.0257
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