Self adaptive inertial relaxed <inline-formula><tex-math id="M1">\begin{document}$ CQ $\end{document}</tex-math></inline-formula> algorithms for solving split feasibility problem with multiple output sets

Guash Haile Taddele; Poom Kumam; Habib ur Rehman; Anteneh Getachew Gebrie

doi:10.3934/jimo.2021172

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Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

Guash Haile Taddele ^1, , Poom Kumam ^2,3,4,, , Habib ur Rehman ^1, and Anteneh Getachew Gebrie ^5,

1.	Department of Mathematics, Faculty of Science, King Mongkut's University of Technology, Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.	Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
3.	Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
4.	Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.	Department of Mathematics, College of Computational and Natural Science, Debre Berhan University, Debre Berhan, PO. Box 445, Ethiopia

^* Corresponding author: E-mail address: poom.kum@kmutt.ac.th (Poom Kumam)

Received April 2021 Revised August 2021 Early access October 2021

Fund Project: The first author is supported by the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi grant No.37/2561

In this paper, we propose two new self-adaptive inertial relaxed $ CQ $ algorithms for solving the split feasibility problem with multiple output sets in the framework of real Hilbert spaces. The proposed algorithms involve computing projections onto half-spaces instead of onto the closed convex sets, and the advantage of the self-adaptive step size introduced in our algorithms is that it does not require the computation of operator norm. We establish and prove weak and strong convergence theorems for the iterative sequences generated by the introduced algorithms for solving the aforementioned problem. Moreover, we apply the new results to solve some other problems. Finally, we present some numerical examples to illustrate the implementation of our algorithms and compared them to some existing results.

Keywords: Split feasibility problem, split feasibility problem with multiple output sets, self-adaptive technique, inertial technique, half-space.

Mathematics Subject Classification: 47H09, 65J15, 65K05, 65K10, 49J52.

Citation: Guash Haile Taddele, Poom Kumam, Habib ur Rehman, Anteneh Getachew Gebrie. Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021172

References:

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[38]	S. Reich and T. M. Tuyen, Iterative methods for solving the generalized split common null point problem in Hilbert spaces, Optimization, 69 (2020), 1013-1038. doi: 10.1080/02331934.2019.1655562.
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[42]	T. Saelii, S. Kesornprom and P. Cholamjiak, A novel relaxed projective method for split feasibility problems, Thai J. Math., 18 (2020), 1359-1373.
[43]	D. R. Sahu, Y. J. Cho, Q. L. Dong, M. R. Kashyap and X. H. Li, Inertial relaxed $CQ$ algorithms for solving a split feasibility problem in Hilbert spaces, Numer. Algorithms, 87 (2021), 1075-1095. doi: 10.1007/s11075-020-00999-2.
[44]	Y. Shehu, Strong convergence theorem for multiple sets split feasibility problems in Banach spaces, Numer. Funct. Anal. Optim., 37 (2016), 1021-1036. doi: 10.1080/01630563.2016.1185614.
[45]	Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510. doi: 10.1007/s11784-017-0435-z.
[46]	Y. Shehu, P. T. Vuong and P. Cholamjiak, A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems, J. Fixed Point Theory Appl., 21 (2019), Paper No. 50, 24 pp. doi: 10.1007/s11784-019-0684-0.
[47]	S. Suantai, N. Pholasa and P. Cholamjiak, 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.
[48]	S. Suantai, N. Pholasa and P. Cholamjiak, Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113 (2019), 1081-1099. doi: 10.1007/s13398-018-0535-7.
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[52]	T. M. Tuyen, N. S. Ha and N. T. T. Thuy, A shrinking projection method for solving the split common null point problem in Banach spaces, Numer. Algorithms, 81 (2019), 813-832. doi: 10.1007/s11075-018-0572-5.
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show all references

References:

[1]	T. Alakoya, L. O. Jolaoso, A. Taiwo and O. Mewomo, Inertial algorithm with self-adaptive step size for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, 1–35.
[2]	F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Analysis, 9 (2001), 3-11. doi: 10.1023/A:1011253113155.
[3]	P. K. Anh, N. T. and V. T. Dung, A new self-adaptive CQ algorithm with an application to the LASSO problem, J. Fixed Point Theory Appl., 20 (2018), Paper No. 142, 19 pp. doi: 10.1007/s11784-018-0620-8.
[4]	J.-P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, vol. 140, Springer-Verlag, Berlin, 1993.
[5]	H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, vol. 408, Springer, 2011. doi: 10.1007/978-1-4419-9467-7.
[6]	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.
[7]	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.
[8]	C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.
[9]	A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, vol. 2057, Springer, 2012.
[10]	A. Cegielski, General method for solving the split common fixed point problem, J. Optim. Theory App., 165 (2015), 385-404. doi: 10.1007/s10957-014-0662-z.
[11]	Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine & Biology, 51 (2006), 2353. doi: 10.1088/0031-9155/51/10/001.
[12]	Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8 (1994), 221-239. doi: 10.1007/BF02142692.
[13]	Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084. doi: 10.1088/0266-5611/21/6/017.
[14]	Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms, 59 (2012), 301-323. doi: 10.1007/s11075-011-9490-5.
[15]	Y. Censor and A. Segal, Iterative projection methods in biomedical inverse problems, Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), 10 (2008), 65-96.
[16]	Y. Censor and A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600.
[17]	Y. Dang, J. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim., 13 (2017), 1383-1394. doi: 10.3934/jimo.2016078.
[18]	Q. L. Dong, H. B. Yuan, Y. J. Cho and Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102. doi: 10.1007/s11590-016-1102-9.
[19]	M. Fukushima, A relaxed projection method for variational inequalities, Math. Programming, 35 (1986), 58-70. doi: 10.1007/BF01589441.
[20]	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.
[21]	A. Gibali, D. T. Mai and et al., A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications, J. Ind. Manag. Optim., 15 (2019), 963-984. doi: 10.3934/jimo.2018080.
[22]	K. Goebel and R. Simeon, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Dekker, 1984.
[23]	S. He and Z. Zhao, Strong convergence of a relaxed CQ algorithm for the split feasibility problem, J. Inequal. Appl., 2013 (2013), 197, 11 pp. doi: 10.1186/1029-242X-2013-197.
[24]	O. S. Iyiola and Y. Shehu, A cyclic iterative method for solving multiple sets split feasibility problems in Banach spaces, Quaest. Math., 39 (2016), 959-975. doi: 10.2989/16073606.2016.1241957.
[25]	S. Kesornprom, N. Pholasa and P. Cholamjiak, A modified CQ algorithm for solving the multiple-sets split feasibility problem and the fixed point problem for nonexpansive mappings, Thai J. Math., 17 (2019), 475-493.
[26]	G. López, V. Martín-Márquez, F. Wang and H.-K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Problems, 28 (2012), 085004, 18 pp. doi: 10.1088/0266-5611/28/8/085004.
[27]	D. A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, 51 (2015), 311-325. doi: 10.1007/s10851-014-0523-2.
[28]	P.-E. Maingé, 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.
[29]	P.-E. Maingé, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set-Valued Anal., 15 (2007), 67-79. doi: 10.1007/s11228-006-0027-3.
[30]	P.-E. Maingé, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236. doi: 10.1016/j.cam.2007.07.021.
[31]	O. T. Mewomo and F. U. Ogbuisi, Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces, Quaest. Math., 41 (2018), 129-148. doi: 10.2989/16073606.2017.1375569.
[32]	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.
[33]	A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2003), 447-454. doi: 10.1016/S0377-0427(02)00906-8.
[34]	Yu. E. Nesterov, A method for solving the convex programming problem with convergence rate o (1/k^ 2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.
[35]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597. doi: 10.1090/S0002-9904-1967-11761-0.
[36]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Mat. Fiz., 4 (1964), 791–803.
[37]	S. Reich, M. T. Truong and T. N. H. Mai, The split feasibility problem with multiple output sets in Hilbert spaces, Optim. Lett., 14 (2020), 2335-2353. doi: 10.1007/s11590-020-01555-6.
[38]	S. Reich and T. M. Tuyen, Iterative methods for solving the generalized split common null point problem in Hilbert spaces, Optimization, 69 (2020), 1013-1038. doi: 10.1080/02331934.2019.1655562.
[39]	S. Reich and T. M. Tuyen, Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 180, 16 pp. doi: 10.1007/s13398-020-00901-8.
[40]	S. Reich and T. M. Tuyen, Two projection algorithms for solving the split common fixed point problem, J. Optim. Theory Appl., 186 (2020), 148-168. doi: 10.1007/s10957-020-01702-0.
[41]	S. Reich, T. M. Tuyen and M. T. N. Ha, An optimization approach to solving the split feasibility problem in Hilbert spaces, Journal of Global Optimization, 79 (2021), 837-852. doi: 10.1007/s10898-020-00964-2.
[42]	T. Saelii, S. Kesornprom and P. Cholamjiak, A novel relaxed projective method for split feasibility problems, Thai J. Math., 18 (2020), 1359-1373.
[43]	D. R. Sahu, Y. J. Cho, Q. L. Dong, M. R. Kashyap and X. H. Li, Inertial relaxed $CQ$ algorithms for solving a split feasibility problem in Hilbert spaces, Numer. Algorithms, 87 (2021), 1075-1095. doi: 10.1007/s11075-020-00999-2.
[44]	Y. Shehu, Strong convergence theorem for multiple sets split feasibility problems in Banach spaces, Numer. Funct. Anal. Optim., 37 (2016), 1021-1036. doi: 10.1080/01630563.2016.1185614.
[45]	Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510. doi: 10.1007/s11784-017-0435-z.
[46]	Y. Shehu, P. T. Vuong and P. Cholamjiak, A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems, J. Fixed Point Theory Appl., 21 (2019), Paper No. 50, 24 pp. doi: 10.1007/s11784-019-0684-0.
[47]	S. Suantai, N. Pholasa and P. Cholamjiak, 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.
[48]	S. Suantai, N. Pholasa and P. Cholamjiak, Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113 (2019), 1081-1099. doi: 10.1007/s13398-018-0535-7.
[49]	A. Taiwo, T. O. Alakoya and O. T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, 86 (2021), 1359-1389. doi: 10.1007/s11075-020-00937-2.
[50]	W. Takahashi, The split feasibility problem and the shrinking projection method in Banach spaces, J. Nonlinear Convex Anal., 16 (2015), 1449-1459.
[51]	W. Takahashi, C.-F. Wen and J.-C. Yao, An implicit algorithm for the split common fixed point problem in Hilbert spaces and applications, Appl. Anal. Optim, 1 (2017), 423-439.
[52]	T. M. Tuyen, N. S. Ha and N. T. T. Thuy, A shrinking projection method for solving the split common null point problem in Banach spaces, Numer. Algorithms, 81 (2019), 813-832. doi: 10.1007/s11075-018-0572-5.
[53]	J. Wang, Y. Hu, C. Li and J.-C. Yao, Linear convergence of CQ algorithms and applications in gene regulatory network inference, Inverse Problems, 33 (2017), 055017, 25 pp. doi: 10.1088/1361-6420/aa6699.
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Figure 1. Comparison of Algorithm 1, Algorithm 3, Scheme (16), Scheme (17) and Scheme (5.1) for different choices of $ \epsilon $

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Figure 2. Comparison of Algorithm 5, Scheme (13), Scheme (14) and Scheme (17) for different choices of initial points

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Table 1. Algorithm 1 and Algorithm 3 for $ \epsilon = 10^{-6} $ and different choices of $ \rho_{1}^{n}, \rho_{2}^{n} $ and $ \theta $

	$ \rho_{1}^{n}=\frac{3n}{4n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{3n}{20n+1}=\rho_{2}^{n} $
	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En
Algorithm 1	22	0.018314	8.41E-07	32	0.025566	7.17E-07	53	0.024086	9.40E-07	74	0.02651	9.69E-07
Algorithm 3	83	0.023672	9.67E-07	91	0.042918	9.61E-07	157	0.028161	9.88E-07	207	0.034091	9.80E-07

	$ \theta=0 $			$ \theta=0.15 $			$ \theta=0.25 $			$ \theta=0.5 $
	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En
Algorithm 1	45	0.025492	8.00E-07	57	0.02797	7.78E-07	43	0.029967	9.76E-07	51	0.02619	8.02E-07
Algorithm 3	111	0.026377	9.79E-07	136	0.030002	9.82E-07	91	0.026963	9.81E-07	84	0.024384	9.82E-07

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	$ \rho_{1}^{n}=\frac{3n}{4n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{3n}{20n+1}=\rho_{2}^{n} $
	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En
Algorithm 1	22	0.018314	8.41E-07	32	0.025566	7.17E-07	53	0.024086	9.40E-07	74	0.02651	9.69E-07
Algorithm 3	83	0.023672	9.67E-07	91	0.042918	9.61E-07	157	0.028161	9.88E-07	207	0.034091	9.80E-07

	$ \theta=0 $			$ \theta=0.15 $			$ \theta=0.25 $			$ \theta=0.5 $
	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En
Algorithm 1	45	0.025492	8.00E-07	57	0.02797	7.78E-07	43	0.029967	9.76E-07	51	0.02619	8.02E-07
Algorithm 3	111	0.026377	9.79E-07	136	0.030002	9.82E-07	91	0.026963	9.81E-07	84	0.024384	9.82E-07

Table 2. Algorithm 1, Algorithm 3, Scheme (16), Scheme (17) and Scheme (5.1) for different choices of $ \epsilon $

		Algorithm 1	Algorithm 3	Scheme (16)	Scheme (17)	Scheme (5.1)
$ \epsilon=10^{-6} $	Iter.(n)	24	75	180	111	75
	CPU(s)	0.01667	0.02255	0.023436	0.037266	0.029002
	$ E_n $	8.25E-07	9.67E-07	9.74E-07	9.82E-07	9.92E-07
$ \epsilon=10^{-7} $	Iter.(n)	30	134	174	282	211
	CPU(s)	0.01962	0.025028	0.025425	0.026567	0.033771
	$ E_n $	6.17E-08	9.83E-08	9.80E-08	9.96E-08	9.99E-08
$ \epsilon=10^{-8} $	Iter.(n)	41	276	470	537	770
	CPU(s)	0.024448	0.029783	0.033593	0.035215	0.038546
	$ E_n $	8.68E-09	9.95E-09	9.84E-09	9.99E-09	9.98E-09
$ \epsilon=10^{-9} $	Iter.(n)	49	479	496	2024	2263
	CPU(s)	0.026591	0.037697	0.036028	0.039359	0.088713
	$ E_n $	6.98E-10	9.93E-10	9.83E-10	1.00E-09	1.00E-09

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		Algorithm 1	Algorithm 3	Scheme (16)	Scheme (17)	Scheme (5.1)
$ \epsilon=10^{-6} $	Iter.(n)	24	75	180	111	75
	CPU(s)	0.01667	0.02255	0.023436	0.037266	0.029002
	$ E_n $	8.25E-07	9.67E-07	9.74E-07	9.82E-07	9.92E-07
$ \epsilon=10^{-7} $	Iter.(n)	30	134	174	282	211
	CPU(s)	0.01962	0.025028	0.025425	0.026567	0.033771
	$ E_n $	6.17E-08	9.83E-08	9.80E-08	9.96E-08	9.99E-08
$ \epsilon=10^{-8} $	Iter.(n)	41	276	470	537	770
	CPU(s)	0.024448	0.029783	0.033593	0.035215	0.038546
	$ E_n $	8.68E-09	9.95E-09	9.84E-09	9.99E-09	9.98E-09
$ \epsilon=10^{-9} $	Iter.(n)	49	479	496	2024	2263
	CPU(s)	0.026591	0.037697	0.036028	0.039359	0.088713
	$ E_n $	6.98E-10	9.93E-10	9.83E-10	1.00E-09	1.00E-09

Table 3. Comparison of Algorithm 5, Scheme (13), Scheme (14) and Scheme (17)

Algorithm 5			Scheme (14)			Scheme (13)			Scheme (17)
Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)
1	1149.360361		1	231.8966545		1	199.2220601		1	640.4875017
2	28.03259245		2	91.40575598		2	69.5163167		2	12.35158962
3	0.811105412		3	32.14832803		3	20.87177641		3	2.434404763
4	0.202894757		4	10.25120406		4	6.548286817		4	0.513146017
5	0.050765569		5	3.137538418		5	2.301982084		5	0.140142486
6	0.012707645		6	1.056033824		6	0.966763557		6	0.083619119
7	0.003183583		7	0.449136255		7	0.488009333		7	0.059127765
8	0.000798736		8	0.228594093		8	0.279757135		8	0.041809171
9	0.000200925		9	0.11848334		9	0.170839653		9	0.029562999
10	5.07839E-05	69.73042	10	0.056872251		10	0.106846934		10	0.020903734
			11	0.024488093		11	0.06722031		11	0.014780829
			12	0.009391695		12	0.042254471		12	0.010451389
			13	0.00322014		13	0.026486762		13	0.007390099
			14	0.001009238		14	0.01655399		14	0.005225502
			15	0.000311098		15	0.010319917		15	0.003694944
			16	0.000109788		16	0.006420625		16	0.002612704
			17	4.93538E-05	140.9432	17	0.003988518		17	0.001847463
						18	0.002474828		18	0.001306366
						19	0.001534285		19	0.000923758
						20	0.000950584		20	0.000653215
						21	0.000588668		21	0.000461913
						22	0.000364417		22	0.00032664
						23	0.000225534		23	0.000230986
						24	0.000139554		24	0.000163347
						25	8.6339E-05	173.0115	25	0.000115517
									26	8.16939E-05	132.0709

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Algorithm 5			Scheme (14)			Scheme (13)			Scheme (17)
Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)
1	1149.360361		1	231.8966545		1	199.2220601		1	640.4875017
2	28.03259245		2	91.40575598		2	69.5163167		2	12.35158962
3	0.811105412		3	32.14832803		3	20.87177641		3	2.434404763
4	0.202894757		4	10.25120406		4	6.548286817		4	0.513146017
5	0.050765569		5	3.137538418		5	2.301982084		5	0.140142486
6	0.012707645		6	1.056033824		6	0.966763557		6	0.083619119
7	0.003183583		7	0.449136255		7	0.488009333		7	0.059127765
8	0.000798736		8	0.228594093		8	0.279757135		8	0.041809171
9	0.000200925		9	0.11848334		9	0.170839653		9	0.029562999
10	5.07839E-05	69.73042	10	0.056872251		10	0.106846934		10	0.020903734
			11	0.024488093		11	0.06722031		11	0.014780829
			12	0.009391695		12	0.042254471		12	0.010451389
			13	0.00322014		13	0.026486762		13	0.007390099
			14	0.001009238		14	0.01655399		14	0.005225502
			15	0.000311098		15	0.010319917		15	0.003694944
			16	0.000109788		16	0.006420625		16	0.002612704
			17	4.93538E-05	140.9432	17	0.003988518		17	0.001847463
						18	0.002474828		18	0.001306366
						19	0.001534285		19	0.000923758
						20	0.000950584		20	0.000653215
						21	0.000588668		21	0.000461913
						22	0.000364417		22	0.00032664
						23	0.000225534		23	0.000230986
						24	0.000139554		24	0.000163347
						25	8.6339E-05	173.0115	25	0.000115517
									26	8.16939E-05	132.0709

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2020 Impact Factor: 1.801

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2020 Impact Factor: 1.801

American Institute of Mathematical Sciences

Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

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American Institute of Mathematical Sciences

Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

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