doi: 10.3934/jimo.2021172
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 and Management Optimization, doi: 10.3934/jimo.2021172
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. ByrneY. CensorA. 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. 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.

[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. 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.

[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.

[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. 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.

[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.

[19]

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

[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.

[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.

[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. 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. 

[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. 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.

[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. 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.

[42]

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

[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.

[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. SuantaiN. 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. SuantaiN. 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. TaiwoT. 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. TakahashiC.-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. TuyenN. 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.

[54]

J. WangY. HuC. K. W. Yu and X. Zhuang, A family of projection gradient methods for solving the multiple-sets split feasibility problem, J. Optim. Theory Appl., 183 (2019), 520-534.  doi: 10.1007/s10957-019-01563-2.

[55]

H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

[56]

H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 105018.  doi: 10.1088/0266-5611/26/10/105018.

[57]

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

[58]

Y. YaoL. LengM. Postolache and X. Zheng, Mann-type iteration method for solving the split common fixed point problem, J. Nonlinear Convex Anal, 18 (2017), 875-882. 

[59]

Y. Yao, M. Postolache and Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 201, 12 pp. doi: 10.1186/1687-1812-2013-201.

[60]

Y. YaoM. Postolache and Z. Zhu, Gradient methods with selection technique for the multiple-sets split feasibility problem, Optimization, 69 (2020), 269-281.  doi: 10.1080/02331934.2019.1602772.

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. ByrneY. CensorA. 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. 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.

[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. 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.

[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.

[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. 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.

[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.

[19]

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

[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.

[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.

[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. 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. 

[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. 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.

[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. 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.

[42]

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

[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.

[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. SuantaiN. 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. SuantaiN. 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. TaiwoT. 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. TakahashiC.-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. TuyenN. 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.

[54]

J. WangY. HuC. K. W. Yu and X. Zhuang, A family of projection gradient methods for solving the multiple-sets split feasibility problem, J. Optim. Theory Appl., 183 (2019), 520-534.  doi: 10.1007/s10957-019-01563-2.

[55]

H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

[56]

H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 105018.  doi: 10.1088/0266-5611/26/10/105018.

[57]

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

[58]

Y. YaoL. LengM. Postolache and X. Zheng, Mann-type iteration method for solving the split common fixed point problem, J. Nonlinear Convex Anal, 18 (2017), 875-882. 

[59]

Y. Yao, M. Postolache and Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 201, 12 pp. doi: 10.1186/1687-1812-2013-201.

[60]

Y. YaoM. Postolache and Z. Zhu, Gradient methods with selection technique for the multiple-sets split feasibility problem, Optimization, 69 (2020), 269-281.  doi: 10.1080/02331934.2019.1602772.

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
[1]

Ya-Zheng Dang, Zhong-Hui Xue, Yan Gao, Jun-Xiang Li. Fast self-adaptive regularization iterative algorithm for solving split feasibility problem. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1555-1569. doi: 10.3934/jimo.2019017

[2]

Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078

[3]

Yazheng Dang, Marcus Ang, Jie Sun. An inertial triple-projection algorithm for solving the split feasibility problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022019

[4]

Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial and Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749

[5]

Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104

[6]

Suthep Suantai, Nattawut Pholasa, Prasit Cholamjiak. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1595-1615. doi: 10.3934/jimo.2018023

[7]

Ya-zheng Dang, Jie Sun, Su Zhang. Double projection algorithms for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2023-2034. doi: 10.3934/jimo.2018135

[8]

Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082

[9]

Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial and Management Optimization, 2022, 18 (1) : 239-265. doi: 10.3934/jimo.2020152

[10]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021046

[11]

Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459

[12]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure and Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957

[13]

Aviv Gibali, Dang Thi Mai, Nguyen The Vinh. A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. Journal of Industrial and Management Optimization, 2019, 15 (2) : 963-984. doi: 10.3934/jimo.2018080

[14]

Jacob Ashiwere Abuchu, Godwin Chidi Ugwunnadi, Ojen Kumar Narain. Inertial Mann-Type iterative method for solving split monotone variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022075

[15]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial and Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255

[16]

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, 2022  doi: 10.3934/naco.2022007

[17]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[18]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[19]

Yong-Jung Kim. A generalization of the moment problem to a complex measure space and an approximation technique using backward moments. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 187-207. doi: 10.3934/dcds.2011.30.187

[20]

Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021160

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (379)
  • HTML views (348)
  • Cited by (0)

[Back to Top]