# American Institute of Mathematical Sciences

October  2018, 14(4): 1595-1615. doi: 10.3934/jimo.2018023

## The modified inertial relaxed CQ algorithm for solving the split feasibility problems

 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand 2 School of Science, University of Phayao, Phayao 56000, Thailand

∗ Corresponding author: prasitch2008@yahoo.com (P. Cholamjiak)

Received  April 2017 Revised  August 2017 Published  October 2018 Early access  January 2018

In this work, we propose a new version of inertial relaxed CQ algorithms for solving the split feasibility problems in the frameworks of Hilbert spaces. We then prove its strong convergence by using a viscosity approximation method under some weakened assumptions. To be more precisely, the computation on the norm of operators and the metric projections is relaxed. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

Citation: Suthep Suantai, Nattawut Pholasa, Prasit Cholamjiak. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1595-1615. doi: 10.3934/jimo.2018023
##### References:
 [1] F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar [2] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin, 1993.  Google Scholar [3] H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problem, SIAM Rev., 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.  Google Scholar [4] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar [5] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar [6] A. Cegielski, General method for solving the split common fixed point problems, J. 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Program., 35 (1986), 58-70.  doi: 10.1007/BF01589441.  Google Scholar [12] S. He and Z. Zhao, Strong convergence of a relaxed CQ algorithm for the split feasibility problem, J. Inqe. Appl. 2013 (2013), p197.  Google Scholar [13] G. López, V. Martin-Marquez, F. H. Wang and H. K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl. 28 (2012), 085004, 18pp. doi: 10.1088/0266-5611/28/8/085004.  Google Scholar [14] D. A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51 (2015), 311-325.  doi: 10.1007/s10851-014-0523-2.  Google Scholar [15] 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 [16] 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 [17] 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 [18] P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.  Google Scholar [19] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615.  Google Scholar [20] 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 [21] Y. Nesterov, A method for solving the convex programming problem with convergence rate (1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.   Google Scholar [22] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, U. S. S. R. Comput. Math. Math. Phys., 4 (1964), 1-17.   Google Scholar [23] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar [24] F. Wang, On the convergence of CQ algorithm with variable steps for the split equality problem, Numer. Algor., 74 (2017), 927-935.  doi: 10.1007/s11075-016-0177-9.  Google Scholar [25] H. K. Xu, A variable Krasonosel'skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl., 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.  Google Scholar [26] H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl. 26 (2010), 105018, 17pp.  Google Scholar [27] H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar [28] Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.  Google Scholar [29] Q. Yang, On variable-set relaxed projection algorithm for variational inequalities, J. Math. Anal. Appl., 302 (2005), 166-179.  doi: 10.1016/j.jmaa.2004.07.048.  Google Scholar

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##### References:
 [1] F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar [2] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin, 1993.  Google Scholar [3] H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problem, SIAM Rev., 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.  Google Scholar [4] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar [5] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar [6] A. Cegielski, General method for solving the split common fixed point problems, J. Optim. Theory Appl., 165 (2015), 385-404.  doi: 10.1007/s10957-014-0662-z.  Google Scholar [7] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in product space, Numer. Algor., 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar [8] Y. Dang and Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Probl. 27 (2011), 015007, 9pp. doi: 10.1088/0266-5611/27/1/015007.  Google Scholar [9] 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.  Google Scholar [10] 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., (2016), 1-16.  doi: 10.1007/s11590-016-1102-9.  Google Scholar [11] M. Fukushima, A relaxed projection method for variational inequalities, Math. Program., 35 (1986), 58-70.  doi: 10.1007/BF01589441.  Google Scholar [12] S. He and Z. Zhao, Strong convergence of a relaxed CQ algorithm for the split feasibility problem, J. Inqe. Appl. 2013 (2013), p197.  Google Scholar [13] G. López, V. Martin-Marquez, F. H. Wang and H. K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl. 28 (2012), 085004, 18pp. doi: 10.1088/0266-5611/28/8/085004.  Google Scholar [14] D. A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51 (2015), 311-325.  doi: 10.1007/s10851-014-0523-2.  Google Scholar [15] 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 [16] 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 [17] 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 [18] P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.  Google Scholar [19] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615.  Google Scholar [20] 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 [21] Y. Nesterov, A method for solving the convex programming problem with convergence rate (1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.   Google Scholar [22] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, U. S. S. R. Comput. Math. Math. Phys., 4 (1964), 1-17.   Google Scholar [23] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar [24] F. Wang, On the convergence of CQ algorithm with variable steps for the split equality problem, Numer. Algor., 74 (2017), 927-935.  doi: 10.1007/s11075-016-0177-9.  Google Scholar [25] H. K. Xu, A variable Krasonosel'skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl., 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.  Google Scholar [26] H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl. 26 (2010), 105018, 17pp.  Google Scholar [27] H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar [28] Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.  Google Scholar [29] Q. Yang, On variable-set relaxed projection algorithm for variational inequalities, J. Math. Anal. Appl., 302 (2005), 166-179.  doi: 10.1016/j.jmaa.2004.07.048.  Google Scholar
Comparison of the iterations of Choice 1 in Example 1
Comparison of the iterations of Choice 2 in Example 1
Comparison of the iterations of Choice 3 in Example 1
Comparison of the iterations of Choice 4 in Example 1
Comparison of the iterations of Choice 1 in Example 2
Comparison of the iterations of Choice 2 in Example 2
Comparison of the iterations of Choice 3 in Example 2
Comparison of the iterations of Choice 4 in Example 2
Error ploting of Choice 1 in Example 1
Error ploting of Choice 2 in Example 1
Error ploting of Choice 3 in Example 1
Error ploting of Choice 4 in Example 1
Algorithm 3.1 with different cases of $\rho_n$ and different choices of $x_0$ and $x_1$
 Case 1 Case 2 Case 3 Case 4 Choice 1 No. of Iter. 11 8 5 4 cpu (Time) $0.003553$ $0.002377$ $0.002195$ $0.002075$ Choice 2 No. of Iter. 7 6 4 4 cpu (Time) $0.002799$ $0.002769$ $0.002357$ $0.002184$ Choice 3 No. of Iter. 12 9 6 4 cpu (Time) $0.003828$ $0.002602$ $0.002401$ $0.002142$ Choice 4 No. of Iter. 27 17 11 9 cpu (Time) $0.007181$ $0.00343$ $0.002612$ $0.002431$ The numerical experiments for each case of $\rho_{n}$ are shown in Figure 1-4, respectively.
 Case 1 Case 2 Case 3 Case 4 Choice 1 No. of Iter. 11 8 5 4 cpu (Time) $0.003553$ $0.002377$ $0.002195$ $0.002075$ Choice 2 No. of Iter. 7 6 4 4 cpu (Time) $0.002799$ $0.002769$ $0.002357$ $0.002184$ Choice 3 No. of Iter. 12 9 6 4 cpu (Time) $0.003828$ $0.002602$ $0.002401$ $0.002142$ Choice 4 No. of Iter. 27 17 11 9 cpu (Time) $0.007181$ $0.00343$ $0.002612$ $0.002431$ The numerical experiments for each case of $\rho_{n}$ are shown in Figure 1-4, respectively.
Algorithm 3.1 with different cases of $\rho_n$ and different choices of $x_0$ and $x_1$
 Case 1 Case 2 Case 3 Case 4 Choice 1 No. of Iter. 19 10 5 5 cpu (Time) $0.005632$ $0.003408$ $0.003223$ $0.002791$ Choice 2 No. of Iter. 18 10 6 6 cpu (Time) $0.00391$ $0.002683$ $0.002447$ $0.002381$ Choice 3 No. of Iter. 19 10 6 6 cpu (Time) $0.004233$ $0.003016$ $0.002601$ $0.002575$ Choice 4 No. of Iter. 13 7 6 6 cpu (Time) $0.004812$ $0.003559$ $0.002922$ $0.002412$ The numerical experiments are shown in Figure 5-8, respectively.
 Case 1 Case 2 Case 3 Case 4 Choice 1 No. of Iter. 19 10 5 5 cpu (Time) $0.005632$ $0.003408$ $0.003223$ $0.002791$ Choice 2 No. of Iter. 18 10 6 6 cpu (Time) $0.00391$ $0.002683$ $0.002447$ $0.002381$ Choice 3 No. of Iter. 19 10 6 6 cpu (Time) $0.004233$ $0.003016$ $0.002601$ $0.002575$ Choice 4 No. of Iter. 13 7 6 6 cpu (Time) $0.004812$ $0.003559$ $0.002922$ $0.002412$ The numerical experiments are shown in Figure 5-8, respectively.
Comparison of MIner-R-Iter, Iner-R-Iter and H-R-Iter in Example 1
 MIner-R-Iter Iner-R-Iter H-R-Iter Choice 1 $u=(0, -1, -5)^T$ No. of Iter. 6 33 223 $x_{0}=(2, 6, -3)^T$ cpu (Time) 0.000737 0.007677 0.064889 $x_{1}=(-2, -1, 8)^T$ Choice 2 $u=(2, 1, 0)^T$ No. of Iter. 4 26 378 $x_{0}=(3, 4, -1)^T$ cpu (Time) 0.000522 0.004861 0.137471 $x_{1}=(-5, -2, 1)^T$ Choice 3 $u=(5, -3, -1)^T$ No. of Iter. 9 29 140 $x_{0}=(2, 1, -1)^T$ cpu (Time) 0.001458 0.005175 0.026824 $x_{1}=(-5, 3, 5)^T$ Choice 4 $u=(-2, -1, 4)^T$ No. of Iter. 9 34 763 $x_{0}=(7.35, 1.75, -3.24)^T$ cpu (Time) 0.001481 0.008058 0.687214 $x_{1}=(-6.34, 0.42, 7.36)^T$ The error plotting of $E_n$ of MIner-R-Iter, Iner-R-Iter and H-R-Iter for each choice in Table 3 is shown in the following figures, respectively.
 MIner-R-Iter Iner-R-Iter H-R-Iter Choice 1 $u=(0, -1, -5)^T$ No. of Iter. 6 33 223 $x_{0}=(2, 6, -3)^T$ cpu (Time) 0.000737 0.007677 0.064889 $x_{1}=(-2, -1, 8)^T$ Choice 2 $u=(2, 1, 0)^T$ No. of Iter. 4 26 378 $x_{0}=(3, 4, -1)^T$ cpu (Time) 0.000522 0.004861 0.137471 $x_{1}=(-5, -2, 1)^T$ Choice 3 $u=(5, -3, -1)^T$ No. of Iter. 9 29 140 $x_{0}=(2, 1, -1)^T$ cpu (Time) 0.001458 0.005175 0.026824 $x_{1}=(-5, 3, 5)^T$ Choice 4 $u=(-2, -1, 4)^T$ No. of Iter. 9 34 763 $x_{0}=(7.35, 1.75, -3.24)^T$ cpu (Time) 0.001481 0.008058 0.687214 $x_{1}=(-6.34, 0.42, 7.36)^T$ The error plotting of $E_n$ of MIner-R-Iter, Iner-R-Iter and H-R-Iter for each choice in Table 3 is shown in the following figures, respectively.
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