# American Institute of Mathematical Sciences

• Previous Article
Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach
• JIMO Home
• This Issue
• Next Article
The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints
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  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, 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. [2] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin, 1993. [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. [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. [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. [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. [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. [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. [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. [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. [11] M. Fukushima, A relaxed projection method for variational inequalities, Math. Program., 35 (1986), 58-70. doi: 10.1007/BF01589441. [12] S. He and Z. Zhao, Strong convergence of a relaxed CQ algorithm for the split feasibility problem, J. Inqe. Appl. 2013 (2013), p197. [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. [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. [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. [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. [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. [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. [19] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55. doi: 10.1006/jmaa.1999.6615. [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. [21] Y. Nesterov, A method for solving the convex programming problem with convergence rate (1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. [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. [23] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898. doi: 10.1137/0314056. [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. [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. [26] H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl. 26 (2010), 105018, 17pp. [27] H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332. [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. [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.

show all references

##### 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. [2] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin, 1993. [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. [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. [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. [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. [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. [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. [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. [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. [11] M. Fukushima, A relaxed projection method for variational inequalities, Math. Program., 35 (1986), 58-70. doi: 10.1007/BF01589441. [12] S. He and Z. Zhao, Strong convergence of a relaxed CQ algorithm for the split feasibility problem, J. Inqe. Appl. 2013 (2013), p197. [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. [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. [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. [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. [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. [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. [19] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55. doi: 10.1006/jmaa.1999.6615. [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. [21] Y. Nesterov, A method for solving the convex programming problem with convergence rate (1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. [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. [23] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898. doi: 10.1137/0314056. [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. [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. [26] H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl. 26 (2010), 105018, 17pp. [27] H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332. [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. [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.
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.
 [1] Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078 [2] Biao Qu, Naihua Xiu. A relaxed extragradient-like method for a class of constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 645-654. doi: 10.3934/jimo.2007.3.645 [3] Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial & Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749 [4] Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95 [5] Jing Zhou, Dejun Chen, Zhenbo Wang, Wenxun Xing. A conic approximation method for the 0-1 quadratic knapsack problem. Journal of Industrial & Management Optimization, 2013, 9 (3) : 531-547. doi: 10.3934/jimo.2013.9.531 [6] Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2017216 [7] Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems & Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019 [8] Andrew E.B. Lim, John B. Moore. A path following algorithm for infinite quadratic programming on a Hilbert space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 653-670. doi: 10.3934/dcds.1998.4.653 [9] Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042 [10] Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219 [11] Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823 [12] Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606 [13] Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 785-796. doi: 10.3934/naco.2012.2.785 [14] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [15] Gaidi Li, Zhen Wang, Dachuan Xu. An approximation algorithm for the $k$-level facility location problem with submodular penalties. Journal of Industrial & Management Optimization, 2012, 8 (3) : 521-529. doi: 10.3934/jimo.2012.8.521 [16] Baolan Yuan, Wanjun Zhang, Yubo Yuan. A Max-Min clustering method for $k$-means algorithm of data clustering. Journal of Industrial & Management Optimization, 2012, 8 (3) : 565-575. doi: 10.3934/jimo.2012.8.565 [17] Qiang Long, Changzhi Wu. A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1279-1296. doi: 10.3934/jimo.2014.10.1279 [18] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [19] Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial & Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003 [20] Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35

2016 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables