# 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  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:

show all references

##### References:
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] 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 [2] Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 [3] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [4] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [5] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [6] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 [7] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [8] Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 [9] Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 [10] Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149 [11] Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 [12] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [13] Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 [14] Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 [15] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [16] Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 [17] Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 [18] Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027 [19] Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 [20] Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190

2019 Impact Factor: 1.366