# American Institute of Mathematical Sciences

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

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

 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.

Citation: Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial and 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. [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. [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. [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. [5] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. A. Krasnosel'skii, Two remarks on the method of successive approximations(in Russion), Uspekhi Mathematicheskikh Nauk, 10 (1955), 123-127. [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. [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. [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. [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. [28] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl, 150 (2011), 275-283. doi: 10.1007/s10957-011-9814-6. [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. [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. [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. [32] T. L. N. Nguyen and Y. Shin, Deterministic sensing matrices in compressive sensing: A survey, Sci. World J., 2013 (2013), 1-6. [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. [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. [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. [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. [37] H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332. [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. [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.

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. [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. [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. [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. [5] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. A. Krasnosel'skii, Two remarks on the method of successive approximations(in Russion), Uspekhi Mathematicheskikh Nauk, 10 (1955), 123-127. [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. [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. [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. [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. [28] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl, 150 (2011), 275-283. doi: 10.1007/s10957-011-9814-6. [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. [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. [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. [32] T. L. N. Nguyen and Y. Shin, Deterministic sensing matrices in compressive sensing: A survey, Sci. World J., 2013 (2013), 1-6. [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. [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. [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. [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. [37] H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332. [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. [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.
different $\lambda$ for Algorithm 3.1 in Example4.1
different initial point x0 for Algorithm 3.1 in Example4.1
different initial point $x_0$ for Algorithm 3.1 in Example4.2
different initial point $x_0$ for Algorithm 3.2 in Example4.3
different $\lambda$ for Algorithm 3.2 in Example4.3
Algorithm 3.3 in Example 4.4
Algorithm 3.3 in Example 4.4
Algorithm 3.3 in Example 4.4
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
 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
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
 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
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
 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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