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

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

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  • 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.

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

    Citation:

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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [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.
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