doi: 10.3934/naco.2022007
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An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces

1. 

Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

2. 

Department of Mathematics, Kano University of Science and Technology, Wudil 713101, Nigeria

3. 

Applied Mathematics for Science and Engineering Research, Unit (AMSERU) Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand

* Corresponding author

Received  July 2021 Revised  March 2022 Early access April 2022

In this paper, we present extension of a class of split variational inequality problem and fixed point problem due to Lohawech et al. (J. Ineq Appl. 358, 2018) to a class of multiple sets split variational inequality problem and common fixed point problem (CMSSVICFP) in Hilbert spaces. Using the Halpern subgradient extragradient theorem of variational inequality problems, we propose a parallel Halpern subgradient extragradient CQ-method with adaptive step-size for solving the CMSSVICFP. We show that a sequence generated by the proposed algorithm converges strongly to the solution of the CMSSVICFP. We give a numerical example and perform some preliminary numerical tests to illustrate the numerical efficiency of our method.

Citation: Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022007
References:
[1]

C. Byrne, Block iterative methods for image reconstruction from projections, Trans. Image Processing, 5 (1996), 96-103. 

[2]

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

[3]

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

[4]

N. Buong, Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces, Numerical Algorithms, 76 (2017), 783-798.  doi: 10.1007/s11075-017-0282-4.

[5]

G. CaiA. GibaliO. S. Iyiola and Y. Shehu, A new double-projection method for solving variational inequalities in Banach spaces, J. Optim. Theory Appl., 178 (2018), 219-239.  doi: 10.1007/s10957-018-1228-2.

[6]

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.

[7]

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., (2020), Paper No. 33, 19 pp. doi: 10.1186/s13660-020-2306-1.

[8]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numerical Algor., 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[9]

Y. Censor and A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600. 

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert spaces, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.

[16]

P. Daniele, F. Giannessi and A. Maugeri, Equilibrium Problems and Variational Models, Kluwer, 2003. doi: 10.1016/B978-008043944-0/50960-4.

[17]

F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementary Problems, Springer-Verlag, New-York, 2003.

[18]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 34 (1963), 138-142. 

[19]

G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser., 7 (1964), 91-140. 

[20]

F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems, Nonsmooth Optimization and Variational Inequality Models, Kluwer, 2004.

[21] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511526152.
[22]

B. Halpern, Fixed point of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.  doi: 10.1090/S0002-9904-1967-11864-0.

[23]

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310.  doi: 10.1007/BF02392210.

[24]

P. D. Khanh and P. T. Vuong, Modified projection method for strongly pseudomonotone variational inequalities, J. Global Optim., 58 (2014), 341-350.  doi: 10.1007/s10898-013-0042-5.

[25] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. 
[26]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Metocon, (1976), 747-756. 

[27]

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.

[28]

P. Lohawech, A. Kaewcharoen and A. Farajzadeh, Algorithms for the common solution of the split variational inequality problems and fixed point problems with applications, J. Ineq. Appl., (2018), Article Number: 358. doi: 10.1186/s13660-018-1942-1.

[29]

G. Lopez, V. Martin-Marquez, F. H. Wang and H. K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., (2012), Article ID 085004. doi: 10.1088/0266-5611/28/8/085004.

[30]

P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.

[31]

Y. V. Malisky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 50 (2014), 271-277.  doi: 10.1007/s10559-014-9614-8.

[32]

S. PenfoldR. ZalasM. CasiraghiM. BrookeY. Censor and R. Schulte, Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy, Phy. Med. Biol., 62 (2017), 3599-3618. 

[33]

L. D. Popov, A modification of the Arrow-Hurwicz method for search of saddle points, Mathematical notes of the Academy of Sciences of the USSR, 28 (1980), 845-848. 

[34]

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

[35]

Y. I. Suleiman, H. urRehan, A. Gibali and P. Kumam, A self-adaptive extragradient CQ-method for a class of bilevel split equilibrium problem with application to Nash Cournot oligopolistic electricity market models, Comput. Applied Math., 39 (2020), Article Number: 293. doi: 10.1007/s40314-020-01338-w.

[36]

W. Takahashi and M. Toyoda, Weak convergence theorem for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560.

[37]

M. Tian and B. N. Jiang, Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Inequal. Appl., (2017), Article Number: 123. doi: 10.1186/s13660-017-1397-9.

[38]

P. Tseng, On linear convergence of iterative methods for the variational inequality problem, J. Comput. Applied Math., 60 (1995), 237-252.  doi: 10.1016/0377-0427(94)00094-H.

[39]

H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659-678.  doi: 10.1023/A:1023073621589.

[40]

H. K. Xu, Averaged mappings and the gradient projection algorithm, J. Optim Theory Appl., 150 (2011), 360-378.  doi: 10.1007/s10957-011-9837-z.

[41]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, (2001), 473–504. doi: 10.1016/S1570-579X(01)80028-8.

[42]

Q. Yang, On variable-step relaxed projection algorithm for variational inequalities, J. Math. Anal. Appl., 302 (2005), 166-79.  doi: 10.1016/j.jmaa.2004.07.048.

[43]

M. Yukawa, K. Slavakis and I. Yamada, Multi-domain adaptive filtering by feasibility splitting, In ICASSP, (2010), 3814–3817.

[44]

W. Zhang, D. Han and Z. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Probl., 25 (2009), Article ID: 115001. doi: 10.1088/0266-5611/25/11/115001.

[45]

T. Y. ZhaoD. Q. Wang and L. C. Ceng, Inertial method for split null point problems with pseudomonotone variational inequality problems, Numer. Funct. Anal. Appl., 42 (2020), 69-90.  doi: 10.1007/s11075-018-0583-2.

show all references

References:
[1]

C. Byrne, Block iterative methods for image reconstruction from projections, Trans. Image Processing, 5 (1996), 96-103. 

[2]

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

[3]

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

[4]

N. Buong, Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces, Numerical Algorithms, 76 (2017), 783-798.  doi: 10.1007/s11075-017-0282-4.

[5]

G. CaiA. GibaliO. S. Iyiola and Y. Shehu, A new double-projection method for solving variational inequalities in Banach spaces, J. Optim. Theory Appl., 178 (2018), 219-239.  doi: 10.1007/s10957-018-1228-2.

[6]

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.

[7]

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., (2020), Paper No. 33, 19 pp. doi: 10.1186/s13660-020-2306-1.

[8]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numerical Algor., 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[9]

Y. Censor and A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600. 

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert spaces, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.

[16]

P. Daniele, F. Giannessi and A. Maugeri, Equilibrium Problems and Variational Models, Kluwer, 2003. doi: 10.1016/B978-008043944-0/50960-4.

[17]

F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementary Problems, Springer-Verlag, New-York, 2003.

[18]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 34 (1963), 138-142. 

[19]

G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser., 7 (1964), 91-140. 

[20]

F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems, Nonsmooth Optimization and Variational Inequality Models, Kluwer, 2004.

[21] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511526152.
[22]

B. Halpern, Fixed point of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.  doi: 10.1090/S0002-9904-1967-11864-0.

[23]

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310.  doi: 10.1007/BF02392210.

[24]

P. D. Khanh and P. T. Vuong, Modified projection method for strongly pseudomonotone variational inequalities, J. Global Optim., 58 (2014), 341-350.  doi: 10.1007/s10898-013-0042-5.

[25] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. 
[26]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Metocon, (1976), 747-756. 

[27]

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.

[28]

P. Lohawech, A. Kaewcharoen and A. Farajzadeh, Algorithms for the common solution of the split variational inequality problems and fixed point problems with applications, J. Ineq. Appl., (2018), Article Number: 358. doi: 10.1186/s13660-018-1942-1.

[29]

G. Lopez, V. Martin-Marquez, F. H. Wang and H. K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., (2012), Article ID 085004. doi: 10.1088/0266-5611/28/8/085004.

[30]

P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.

[31]

Y. V. Malisky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 50 (2014), 271-277.  doi: 10.1007/s10559-014-9614-8.

[32]

S. PenfoldR. ZalasM. CasiraghiM. BrookeY. Censor and R. Schulte, Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy, Phy. Med. Biol., 62 (2017), 3599-3618. 

[33]

L. D. Popov, A modification of the Arrow-Hurwicz method for search of saddle points, Mathematical notes of the Academy of Sciences of the USSR, 28 (1980), 845-848. 

[34]

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

[35]

Y. I. Suleiman, H. urRehan, A. Gibali and P. Kumam, A self-adaptive extragradient CQ-method for a class of bilevel split equilibrium problem with application to Nash Cournot oligopolistic electricity market models, Comput. Applied Math., 39 (2020), Article Number: 293. doi: 10.1007/s40314-020-01338-w.

[36]

W. Takahashi and M. Toyoda, Weak convergence theorem for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560.

[37]

M. Tian and B. N. Jiang, Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Inequal. Appl., (2017), Article Number: 123. doi: 10.1186/s13660-017-1397-9.

[38]

P. Tseng, On linear convergence of iterative methods for the variational inequality problem, J. Comput. Applied Math., 60 (1995), 237-252.  doi: 10.1016/0377-0427(94)00094-H.

[39]

H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659-678.  doi: 10.1023/A:1023073621589.

[40]

H. K. Xu, Averaged mappings and the gradient projection algorithm, J. Optim Theory Appl., 150 (2011), 360-378.  doi: 10.1007/s10957-011-9837-z.

[41]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, (2001), 473–504. doi: 10.1016/S1570-579X(01)80028-8.

[42]

Q. Yang, On variable-step relaxed projection algorithm for variational inequalities, J. Math. Anal. Appl., 302 (2005), 166-79.  doi: 10.1016/j.jmaa.2004.07.048.

[43]

M. Yukawa, K. Slavakis and I. Yamada, Multi-domain adaptive filtering by feasibility splitting, In ICASSP, (2010), 3814–3817.

[44]

W. Zhang, D. Han and Z. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Probl., 25 (2009), Article ID: 115001. doi: 10.1088/0266-5611/25/11/115001.

[45]

T. Y. ZhaoD. Q. Wang and L. C. Ceng, Inertial method for split null point problems with pseudomonotone variational inequality problems, Numer. Funct. Anal. Appl., 42 (2020), 69-90.  doi: 10.1007/s11075-018-0583-2.

Figure 1.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 5) $ and TOL = $ 10^{-5} $
Figure 2.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 5) $ and TOL = $ 10^{-5} $
Figure 3.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 10) $ and TOL = $ 10^{-5} $
Figure 4.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 10) $ and TOL = $ 10^{-5} $
Figure 5.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (10; 10) $ and TOL = $ 10^{-5} $
Figure 6.  Numerical illustration of Algorithm 3.1 with S(N; M) = $ (10; 10) $ and TOL = $ 10^{-5} $
Figure 7.  Numerical illustration of Algorithm 3.1 with different values of $ \kappa_{n} = \frac{1}{k(n+2)} $
Figure 8.  Numerical illustration of Algorithm 3.1 with different values of $ \kappa_{n} = \frac{1}{k(n+2)} $
Figure 9.  Numerical illustration of Algorithm 3.1 with different values of $ \rho_{n} = k $
Figure 10.  Numerical illustration of Algorithm 3.1 with different values of $ \rho_{n} = k $
Figure 11.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (1, 3) $ and TOL = $ 10^{-4} $
Figure 12.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (1, 3) $ and TOL = $ 10^{-4} $
Figure 13.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (3, 4) $ and TOL = $ 10^{-4} $
Figure 14.  Numerical illustration of Algorithm 3.1 with $ x_{1} = (3, 4) $ and TOL = $ 10^{-4} $
Table 1.  Numerical data for Experiment 1
N.P S(N; M) TOL. CPU(s) ITER.
$ 1 $ (5; 5) $ 10^{-5} $ 4.66201370000000 68
$ 2 $ (5; 10) $ 10^{-5} $ 8.59728030000000 76
$ 3 $ (10; 10) $ 10^{-5} $ 9.57778720000000 92
$ 4 $ (10; 20) $ 10^{-5} $ 13.4638439200000 108
$ 5 $ (20; 30) $ 10^{-5} $ 20.5758686930000 123
N.P S(N; M) TOL. CPU(s) ITER.
$ 1 $ (5; 5) $ 10^{-5} $ 4.66201370000000 68
$ 2 $ (5; 10) $ 10^{-5} $ 8.59728030000000 76
$ 3 $ (10; 10) $ 10^{-5} $ 9.57778720000000 92
$ 4 $ (10; 20) $ 10^{-5} $ 13.4638439200000 108
$ 5 $ (20; 30) $ 10^{-5} $ 20.5758686930000 123
Table 2.  Numerical data for Experiment 2
k S(N; M) TOL. CPU(s) ITER.
$ 1 $ (5; 10) $ 10^{-5} $ 6.29621610000000 80
$ 3 $ (5; 10) $ 10^{-5} $ 7.10547680000000 92
$ 10 $ (5; 10) $ 10^{-5} $ 10.2573273000000 113
$ 20 $ (5; 10) $ 10^{-5} $ 18.1375856000000 155
$ 50 $ (5; 10) $ 10^{-5} $ 30.5698815000000 226
k S(N; M) TOL. CPU(s) ITER.
$ 1 $ (5; 10) $ 10^{-5} $ 6.29621610000000 80
$ 3 $ (5; 10) $ 10^{-5} $ 7.10547680000000 92
$ 10 $ (5; 10) $ 10^{-5} $ 10.2573273000000 113
$ 20 $ (5; 10) $ 10^{-5} $ 18.1375856000000 155
$ 50 $ (5; 10) $ 10^{-5} $ 30.5698815000000 226
Table 3.  Numerical data for Experiment 3
k S(N; M) TOL. CPU(s) ITER.
$ 0.15 $ (5; 10) $ 10^{-5} $ 18.6105856000000 208
$ 0.65 $ (5; 10) $ 10^{-5} $ 16.1703019000000 167
$ 1.15 $ (5; 10) $ 10^{-5} $ 11.3304408000000 128
$ 1.64 $ (5; 10) $ 10^{-5} $ 9.50234320000000 106
$ 1.89 $ (5; 10) $ 10^{-5} $ 6.33439070000000 67
k S(N; M) TOL. CPU(s) ITER.
$ 0.15 $ (5; 10) $ 10^{-5} $ 18.6105856000000 208
$ 0.65 $ (5; 10) $ 10^{-5} $ 16.1703019000000 167
$ 1.15 $ (5; 10) $ 10^{-5} $ 11.3304408000000 128
$ 1.64 $ (5; 10) $ 10^{-5} $ 9.50234320000000 106
$ 1.89 $ (5; 10) $ 10^{-5} $ 6.33439070000000 67
Table 4.  Numerical data for Example 4.2
SAlgorithm Algorithm 3.1
$ x_{1} $ CPU(s) ITER. CPU(s) ITER.
$ (1, 3) $ 10.7807425000000 577 3.90895790000000 282
$ (3, 4) $ 40.4251408000000 1926 8.90772100000000 507
SAlgorithm Algorithm 3.1
$ x_{1} $ CPU(s) ITER. CPU(s) ITER.
$ (1, 3) $ 10.7807425000000 577 3.90895790000000 282
$ (3, 4) $ 40.4251408000000 1926 8.90772100000000 507
[1]

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 and Optimization, 2021  doi: 10.3934/naco.2021037

[2]

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