doi: 10.3934/jimo.2021172
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Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

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.

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 & Management Optimization, doi: 10.3934/jimo.2021172
References:
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Y. CensorT. ElfvingN. 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.  Google Scholar

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

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Y. DangJ. 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.  Google Scholar

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Q. L. DongH. B. YuanY. 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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J.-P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, vol. 140, Springer-Verlag, Berlin, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, vol. 2057, Springer, 2012.  Google Scholar

[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.  Google Scholar

[11]

Y. CensorT. BortfeldB. 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.  Google Scholar

[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.  Google Scholar

[13]

Y. CensorT. ElfvingN. 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.  Google Scholar

[14]

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

[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.   Google Scholar

[16]

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

[17]

Y. DangJ. 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.  Google Scholar

[18]

Q. L. DongH. B. YuanY. 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.  Google Scholar

[19]

M. Fukushima, A relaxed projection method for variational inequalities, Math. Programming, 35 (1986), 58-70.  doi: 10.1007/BF01589441.  Google Scholar

[20]

A. GibaliL.-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

[21]

A. GibaliD. 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.  Google Scholar

[22]

K. Goebel and R. Simeon, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Dekker, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. KesornpromN. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[36]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Mat. Fiz., 4 (1964), 791–803.  Google Scholar

[37]

S. ReichM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

S. ReichT. 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.  Google Scholar

[42]

T. SaeliiS. Kesornprom and P. Cholamjiak, A novel relaxed projective method for split feasibility problems, Thai J. Math., 18 (2020), 1359-1373.   Google Scholar

[43]

D. R. SahuY. J. ChoQ. L. DongM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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