American Institute of Mathematical Sciences

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March  2020, 16(2): 945-964. doi: 10.3934/jimo.2018187

Convergence analysis of a new iterative algorithm for solving split variational inclusion problems

Yan Tang ,

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing 400067, China

* Corresponding author: Yan Tang

Received  May 2018 Revised  August 2018 Published  March 2020 Early access  January 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China (11471059) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 1706154) and the Research Project of Chongqing Technology and Business University (KFJJ2017069)

The split variational inclusion problem (SVIP) has been extensively studied and applied in real-world problems such as intensity-modulated radiation therapy (IMRT) and in sensor networks and in computerized tomography and data compression. Inspired by the works of L$\acute{o}$pez et al.$[24]$, Byrne et al.[10] and Sitthithakerngkiet et al.[34], as well as of Moudafi and Thukur[29], we propose a self-adaptive step size algorithm for solving split variational inclusion problem (SVIP) without the prior knowledge of the operator norms. Under more mild conditions we obtain weak convergence of the proposed algorithm. We also construct a self-adaptive step size two-step iterative algorithm which converges strongly to the minimum-norm element of the solution of the SVIP. Finally, the performances and computational examples are presented and a comparison with related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

Keywords: Split variational inclusion problem, convex minimization problems, self-adaptive step size, set-valued, strong convergence.
Mathematics Subject Classification: Primary: 65K05; 65K10; 90C25; Secondary: 49J53; 49M37.
Citation: Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187
References:
[1]

Q. H. Ansari and A. Rehan, Iterative methods for generalized split feasibility problems in Banach spaces, Carpathian Journal of Mathematics, 33 (2017), 9-26.   Google Scholar

[2]

Q. H. Ansari and A. Rehan, Split feasibility and fixed point problems, in Nonlinear Analysis: Approximation Theory, Optimization and Applications, Edited by Q.H. Ansari, Birkhäuser, Springer, New Delhi, Heidelberg, New York, Dordrecht, London, (2014), 281-322.  Google Scholar

[3]

Q. H. Ansari, A. Rehan and C. F. Wen, Implicit and explicit algorithms for split common fixed point problems, Journal of Nonlinear and Convex Analysis, 17 (2016), 1381-1397.  Google Scholar

[4]

Q. H. Ansari, A. Rehan and J. C. Yao, Split feasibility and fixed point problems for asymptotically k-strict pseudo-contractive mappings in intermediate sense, Fixed Point Theory, 18 (2017), 57-68.  doi: 10.24193/fpt-ro.2017.1.06.  Google Scholar

[5]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6.  Google Scholar

[6]

J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Hou Journal of Mathematics, 4 (1978), 1-9.  Google Scholar

[7]

F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bulletin of the American Mathematical Society, 72 (1966), 571-575.  doi: 10.1090/S0002-9904-1966-11544-6.  Google Scholar

[8]

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

[9]

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

[10]

C. Byrne, Y. Censor, A. Gibali and S. Reich, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear ConvexAnal, 13 (2012), 759-775.   Google Scholar

[11]

L. C. Ceng, Q. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642. doi: 10.1016/j.camwa.2011.12.074.  Google Scholar

[12]

L. C. Ceng, Q. H. Ansari and J. C. Yao, Mann type iterative methods for finding a common solution of split feasibility and fixed point problems, Positivity, 16 (2012), 471-495. doi: 10.1007/s11117-012-0174-8.  Google Scholar

[13]

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

[14]

Y. Censor, T.Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phy. Med. Biol., 51 (2003), 2353-2365. Google Scholar

[15]

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

[16]

Y. Dang and Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (2011), ID 015007, 9pp. doi: 10.1088/0266-5611/27/1/015007.  Google Scholar

[17]

J. Deepho and P. Kumam, The hybrid steepest descent method for split variational inclusion and constrained convex minimization problems, Abstract and Applied Analysis, 2014 (2014), Article ID 365203, 13pp. doi: 10.1155/2014/365203.  Google Scholar

[18]

A. Gibali, L. W. Liu and Y. C. Tang, Note on the modified relaxation CQ algorithm for the split feasibility problem, Optim. Lett., 12 (2018), 817-830. doi: 10.1007/s11590-017-1148-3.  Google Scholar

[19]

A. Gibali, D. T. Mai and T. V. Nguyen, A new relaxed CQ algorithm for solving Split Feasibility Problems in Hilbert spaces and its applications, J. Indus. Manag. Optim., 2018 (2018), 1-25. Google Scholar

[20]

B. Halpern, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society, 73 (1967), 957-961. doi: 10.1090/S0002-9904-1967-11864-0.  Google Scholar

[21]

S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in Banach space, Processing of the American Mathematical Society, 59 (1976), 65-71.  doi: 10.1090/S0002-9939-1976-0412909-X.  Google Scholar

[22]

K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett., 8 (2014), 1113-1124.  doi: 10.1007/s11590-013-0629-2.  Google Scholar

[23]

M. A. Krasnosel'skii, Two remarks on the method of successive approximations(in Russion), Uspekhi Mathematicheskikh Nauk, 10 (1955), 123-127.   Google Scholar

[24]

G. López, V. Martin-Marquez and H. K. Xu, Solving the split feasibilty problem without prior knowledge of matrix norms, Inverse Problems, 28 (2012), ID085004, 18pp. doi: 10.1088/0266-5611/28/8/085004.  Google Scholar

[25]

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.  Google Scholar

[26]

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

[27]

W. R. Mann, Mean value methods in iteration, Processing of the American Mathematical Society, 4 (1953), 506-510. doi: 10.1090/S0002-9939-1953-0054846-3.  Google Scholar

[28]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl, 150 (2011), 275-283. doi: 10.1007/s10957-011-9814-6.  Google Scholar

[29]

A. Moudafi and B. S. Thakur, Solving proximal split feasibilty problem without prior knowledge of matrix norms, Optimization Letters, 8 (2014), 2099-2110. doi: 10.1007/s11590-013-0708-4.  Google Scholar

[30]

A. Moudafi and A. Gibali, $l_1$-$l_2$ regularization of split feasibility problems, Numer. Algor., 78 (2018), 739-757. doi: 10.1007/s11075-017-0398-6.  Google Scholar

[31]

A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 055007, 6 pp. doi: 10.1088/0266-5611/26/5/055007.  Google Scholar

[32]

T. L. N. Nguyen and Y. Shin, Deterministic sensing matrices in compressive sensing: A survey, Sci. World J., 2013 (2013), 1-6. Google Scholar

[33]

Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibiliy problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2843-2510. doi: 10.1007/s11784-017-0435-z.  Google Scholar

[34]

K. Sitthithakerngkiet, J. Deepho and P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution o split variational inclusion and optimization problems, Numer. Algorithms, 79 (2018), 801-824. doi: 10.1007/s11075-017-0462-2.  Google Scholar

[35]

K. Sitthithakerngkiet, J. Deepho and P. Kumam, A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems, Applied Mathematics and Computation, 250 (2015), 986-1001.  doi: 10.1016/j.amc.2014.10.130.  Google Scholar

[36]

T. Suzuki, A sufficient and necesssary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Processing of the American Mathematical Society, 135 (2007), 99-106. doi: 10.1090/S0002-9939-06-08435-8.  Google Scholar

[37]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.  Google Scholar

[38]

Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266. doi: 10.1088/0266-5611/20/4/014.  Google Scholar

[39]

L. Yang and F. H. Zhao, General split variational inclusion problem in hilbert spaces, Abstract and Applied Analysis, 2014 (2014), Article ID 816035, 8pp. doi: 10.1155/2014/816035.  Google Scholar

show all references

References:
[1]

Q. H. Ansari and A. Rehan, Iterative methods for generalized split feasibility problems in Banach spaces, Carpathian Journal of Mathematics, 33 (2017), 9-26.   Google Scholar

[2]

Q. H. Ansari and A. Rehan, Split feasibility and fixed point problems, in Nonlinear Analysis: Approximation Theory, Optimization and Applications, Edited by Q.H. Ansari, Birkhäuser, Springer, New Delhi, Heidelberg, New York, Dordrecht, London, (2014), 281-322.  Google Scholar

[3]

Q. H. Ansari, A. Rehan and C. F. Wen, Implicit and explicit algorithms for split common fixed point problems, Journal of Nonlinear and Convex Analysis, 17 (2016), 1381-1397.  Google Scholar

[4]

Q. H. Ansari, A. Rehan and J. C. Yao, Split feasibility and fixed point problems for asymptotically k-strict pseudo-contractive mappings in intermediate sense, Fixed Point Theory, 18 (2017), 57-68.  doi: 10.24193/fpt-ro.2017.1.06.  Google Scholar

[5]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6.  Google Scholar

[6]

J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Hou Journal of Mathematics, 4 (1978), 1-9.  Google Scholar

[7]

F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bulletin of the American Mathematical Society, 72 (1966), 571-575.  doi: 10.1090/S0002-9904-1966-11544-6.  Google Scholar

[8]

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

[9]

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

[10]

C. Byrne, Y. Censor, A. Gibali and S. Reich, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear ConvexAnal, 13 (2012), 759-775.   Google Scholar

[11]

L. C. Ceng, Q. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642. doi: 10.1016/j.camwa.2011.12.074.  Google Scholar

[12]

L. C. Ceng, Q. H. Ansari and J. C. Yao, Mann type iterative methods for finding a common solution of split feasibility and fixed point problems, Positivity, 16 (2012), 471-495. doi: 10.1007/s11117-012-0174-8.  Google Scholar

[13]

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

[14]

Y. Censor, T.Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phy. Med. Biol., 51 (2003), 2353-2365. Google Scholar

[15]

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

[16]

Y. Dang and Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (2011), ID 015007, 9pp. doi: 10.1088/0266-5611/27/1/015007.  Google Scholar

[17]

J. Deepho and P. Kumam, The hybrid steepest descent method for split variational inclusion and constrained convex minimization problems, Abstract and Applied Analysis, 2014 (2014), Article ID 365203, 13pp. doi: 10.1155/2014/365203.  Google Scholar

[18]

A. Gibali, L. W. Liu and Y. C. Tang, Note on the modified relaxation CQ algorithm for the split feasibility problem, Optim. Lett., 12 (2018), 817-830. doi: 10.1007/s11590-017-1148-3.  Google Scholar

[19]

A. Gibali, D. T. Mai and T. V. Nguyen, A new relaxed CQ algorithm for solving Split Feasibility Problems in Hilbert spaces and its applications, J. Indus. Manag. Optim., 2018 (2018), 1-25. Google Scholar

[20]

B. Halpern, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society, 73 (1967), 957-961. doi: 10.1090/S0002-9904-1967-11864-0.  Google Scholar

[21]

S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in Banach space, Processing of the American Mathematical Society, 59 (1976), 65-71.  doi: 10.1090/S0002-9939-1976-0412909-X.  Google Scholar

[22]

K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett., 8 (2014), 1113-1124.  doi: 10.1007/s11590-013-0629-2.  Google Scholar

[23]

M. A. Krasnosel'skii, Two remarks on the method of successive approximations(in Russion), Uspekhi Mathematicheskikh Nauk, 10 (1955), 123-127.   Google Scholar

[24]

G. López, V. Martin-Marquez and H. K. Xu, Solving the split feasibilty problem without prior knowledge of matrix norms, Inverse Problems, 28 (2012), ID085004, 18pp. doi: 10.1088/0266-5611/28/8/085004.  Google Scholar

[25]

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.  Google Scholar

[26]

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

[27]

W. R. Mann, Mean value methods in iteration, Processing of the American Mathematical Society, 4 (1953), 506-510. doi: 10.1090/S0002-9939-1953-0054846-3.  Google Scholar

[28]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl, 150 (2011), 275-283. doi: 10.1007/s10957-011-9814-6.  Google Scholar

[29]

A. Moudafi and B. S. Thakur, Solving proximal split feasibilty problem without prior knowledge of matrix norms, Optimization Letters, 8 (2014), 2099-2110. doi: 10.1007/s11590-013-0708-4.  Google Scholar

[30]

A. Moudafi and A. Gibali, $l_1$-$l_2$ regularization of split feasibility problems, Numer. Algor., 78 (2018), 739-757. doi: 10.1007/s11075-017-0398-6.  Google Scholar

[31]

A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 055007, 6 pp. doi: 10.1088/0266-5611/26/5/055007.  Google Scholar

[32]

T. L. N. Nguyen and Y. Shin, Deterministic sensing matrices in compressive sensing: A survey, Sci. World J., 2013 (2013), 1-6. Google Scholar

[33]

Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibiliy problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2843-2510. doi: 10.1007/s11784-017-0435-z.  Google Scholar

[34]

K. Sitthithakerngkiet, J. Deepho and P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution o split variational inclusion and optimization problems, Numer. Algorithms, 79 (2018), 801-824. doi: 10.1007/s11075-017-0462-2.  Google Scholar

[35]

K. Sitthithakerngkiet, J. Deepho and P. Kumam, A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems, Applied Mathematics and Computation, 250 (2015), 986-1001.  doi: 10.1016/j.amc.2014.10.130.  Google Scholar

[36]

T. Suzuki, A sufficient and necesssary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Processing of the American Mathematical Society, 135 (2007), 99-106. doi: 10.1090/S0002-9939-06-08435-8.  Google Scholar

[37]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.  Google Scholar

[38]

Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266. doi: 10.1088/0266-5611/20/4/014.  Google Scholar

[39]

L. Yang and F. H. Zhao, General split variational inclusion problem in hilbert spaces, Abstract and Applied Analysis, 2014 (2014), Article ID 816035, 8pp. doi: 10.1155/2014/816035.  Google Scholar

Figure 1.  different $ \lambda $ for Algorithm 3.1 in Example4.1
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Figure 2.  different initial point x0 for Algorithm 3.1 in Example4.1
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Figure 3.  different initial point $ x_0 $ for Algorithm 3.1 in Example4.2
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Figure 4.  different initial point $ x_0 $ for Algorithm 3.2 in Example4.3
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Figure 5.  different $ \lambda $ for Algorithm 3.2 in Example4.3
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Figure 6.  Algorithm 3.3 in Example 4.4
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Figure 7.  Algorithm 3.3 in Example 4.4
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Figure 8.  Algorithm 3.3 in Example 4.4
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Table 1.  The Convergence of Sequence in Alg 3.2
n $ x_n $ $ \|x_n\| $
0 (1, -1, 2) 2.4490
1 (0.3331, -0.4494, -1.31740) 1.4313
2 (0.1301, -0.2359, -1.0027) 1.0383
3 (0.0555, -0.1328, -0.8102) 0.8229
4 (0.0251, -0.0765, -0.6615) 0.6664
5 (0.0120, -0.0443, -0.5355) 0.5374
...
10 (0.0010, -0.0035, -0.1571) 0.1572
20 (0.0001, -0.0001, -0.0106) 0.0106
30 (0.0000, 0.0000, -0.00072) 7.1865e-04
36 (0.0000, 0.0000, -0.0001) 1.4345e-04
37 (0.0000, 0.0000, -0.0001) 1.097e-04
38 (0.0000, 0.0000, 0.0000) 0.0000
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n $ x_n $ $ \|x_n\| $
0 (1, -1, 2) 2.4490
1 (0.3331, -0.4494, -1.31740) 1.4313
2 (0.1301, -0.2359, -1.0027) 1.0383
3 (0.0555, -0.1328, -0.8102) 0.8229
4 (0.0251, -0.0765, -0.6615) 0.6664
5 (0.0120, -0.0443, -0.5355) 0.5374
...
10 (0.0010, -0.0035, -0.1571) 0.1572
20 (0.0001, -0.0001, -0.0106) 0.0106
30 (0.0000, 0.0000, -0.00072) 7.1865e-04
36 (0.0000, 0.0000, -0.0001) 1.4345e-04
37 (0.0000, 0.0000, -0.0001) 1.097e-04
38 (0.0000, 0.0000, 0.0000) 0.0000
Table 2.  The Convergence of Sequence Alg3.3
n $ x_n $ $ y_n $ $ \|x_n\| $ $ \|y_n\| $
0 (13, -12, 25) (4.4303, -5.5150, 17.4474) 30.6268 18.8270
1 (2.2152, -2.7575, 8.7237) (0.9781, -1.4819, 6.3531) 9.4135 6.5966
2 (0.4328, -0.7128, 3.5425) (0.2146, -0.4159, 2.6836) 3.6393 2.7241
3 (0.0883, -0.1959, 1.5783) (0.0464, -0.1202, 1.2365) 1.5928 1.2432
4 (0.0180, -0.0560, 0.7595) (0.0095, -0.0353, 0.6112) 0.7617 0.6123
5 (0.0033, -0.0162, 0.3900) (0.0015, -0.0102, 0.3204) 0.3904 0.3206
...
10 (-0.0001, 0.0002, 0.0248) (-0.0001, 0.0002, 0.0213) 0.0248 0.0213
15 (0.0000, 0.0000, 0.0037) (0.0000, 0.0000, 0.0032) 0.0037 0.0032
18 (0.0000, 0.0000, 0.00093) (0.0000, 0.0000, 0.00081) 9.2969e-04 8.0753e-04
19 (0.0000, 0.0000, 0.00059) (0.0000, 0.0000, 0.00052) 5.9324e-04 5.1575e-04
20 (0.0000, 0.0000, 0.00038) (0.0000, 0.0000, 0.0003) 3.7973e-04 3.304e-04
21 (0.0000, 0.0000, 0.0000) (0.0000, 0.0000, 0.0000) 0 0
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n $ x_n $ $ y_n $ $ \|x_n\| $ $ \|y_n\| $
0 (13, -12, 25) (4.4303, -5.5150, 17.4474) 30.6268 18.8270
1 (2.2152, -2.7575, 8.7237) (0.9781, -1.4819, 6.3531) 9.4135 6.5966
2 (0.4328, -0.7128, 3.5425) (0.2146, -0.4159, 2.6836) 3.6393 2.7241
3 (0.0883, -0.1959, 1.5783) (0.0464, -0.1202, 1.2365) 1.5928 1.2432
4 (0.0180, -0.0560, 0.7595) (0.0095, -0.0353, 0.6112) 0.7617 0.6123
5 (0.0033, -0.0162, 0.3900) (0.0015, -0.0102, 0.3204) 0.3904 0.3206
...
10 (-0.0001, 0.0002, 0.0248) (-0.0001, 0.0002, 0.0213) 0.0248 0.0213
15 (0.0000, 0.0000, 0.0037) (0.0000, 0.0000, 0.0032) 0.0037 0.0032
18 (0.0000, 0.0000, 0.00093) (0.0000, 0.0000, 0.00081) 9.2969e-04 8.0753e-04
19 (0.0000, 0.0000, 0.00059) (0.0000, 0.0000, 0.00052) 5.9324e-04 5.1575e-04
20 (0.0000, 0.0000, 0.00038) (0.0000, 0.0000, 0.0003) 3.7973e-04 3.304e-04
21 (0.0000, 0.0000, 0.0000) (0.0000, 0.0000, 0.0000) 0 0
Table 3.  Comparison Alg3.3 with Other Algorithms
DOL Method Step Size Iter CPU Time $ \frac{\|z-x_n\|}{\|x_0-x_{n+1}\|} $
$ 10^{-4} $ Algorithm3.3 $ \tau_n $ 20 0.0698 1.2399e-05
Sitthithakerngkiet et al.[34] 0.001 23 0.0763 7.4893e-06
Byrne et al.[10] 0.001 20 0.1187 4.7322e-05
$ 10^{-5} $ Algorithm3.3 $ \tau_n $ 25 0.0761 1.3826e-06
Sitthithakerngkiet et al.[34] 0.001 219 0.0924 5.2309e-07
Byrne et al.[10] 0.001 24 0.1141 3.7345e-06
$ 10^{-6} $ Algorithm3.3 $ \tau_n $ 31 0.081 1.0539e-07
Sitthithakerngkiet et al.[34] 0.001 2172 0.162 5.2141e-08
Byrne et al.[10] 0.001 27 0.1154 5.5604e-07
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DOL Method Step Size Iter CPU Time $ \frac{\|z-x_n\|}{\|x_0-x_{n+1}\|} $
$ 10^{-4} $ Algorithm3.3 $ \tau_n $ 20 0.0698 1.2399e-05
Sitthithakerngkiet et al.[34] 0.001 23 0.0763 7.4893e-06
Byrne et al.[10] 0.001 20 0.1187 4.7322e-05
$ 10^{-5} $ Algorithm3.3 $ \tau_n $ 25 0.0761 1.3826e-06
Sitthithakerngkiet et al.[34] 0.001 219 0.0924 5.2309e-07
Byrne et al.[10] 0.001 24 0.1141 3.7345e-06
$ 10^{-6} $ Algorithm3.3 $ \tau_n $ 31 0.081 1.0539e-07
Sitthithakerngkiet et al.[34] 0.001 2172 0.162 5.2141e-08
Byrne et al.[10] 0.001 27 0.1154 5.5604e-07
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