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doi: 10.3934/jimo.2022019
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An inertial triple-projection algorithm for solving the split feasibility problem

 1 Department of Management, University of Shanghai for Science and Technology, Shanghai PRC 200093 2 Lee Kong Chian School of Business, Singapore Management University, Singapore 178899 3 School of Electric Engineering, Computing, and Mathematical Science, Curtin University, Bentley, Australia 6102 and, School of Business, National University of Singapore, Singapore 119245

*Corresponding author: Yazheng Dang

Received  August 2021 Revised  December 2021 Early access February 2022

This paper proposes a new inertial triple-projection algorithm for solving the split feasibility problem. The process of projections is divided into three parts. Each part adopts a different variable stepsize to obtain its projection point, which is different from the existing extragradient methods. Flexible rules are employed for selecting the stepsizes and the inertial technique is used for improving the convergence. Convergence results are proven. Numerical experiments show that the proposed method converges more quickly than the general CQ algorithm.

Citation: Yazheng Dang, Marcus Ang, Jie Sun. An inertial triple-projection algorithm for solving the split feasibility problem. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022019
References:
 [1] P. K. Anh, D. V. Thong and N. T. Vinh, Improved inertial extragradient methods for solving pseudo-monotone variational inequalities, Optimization, 14 (2020), 1157-1175. [2] N. Buong, P. Hoai and K. T. Binh, Iterative regularization methods for the multiple-sets split feasibility problem in hilbert spaces, Acta Appl. Math., 165 (2020), 183-197.  doi: 10.1007/s10440-019-00249-1. [3] C. Byrne, An unified treatment of some iterative algorithm in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006. [4] 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. [5] Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Math. Program., 42 (1998), 307-325.  doi: 10.1007/BF01589408. [6] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692. [7] P. Cholamjiak, S. Suantai and Su antai S., A new CQ algorithm for solving split feasibility problems in Hilbert spaces, Bull. Malay. Math. Sci. Soc., 42 (2019), 2517-2534.  doi: 10.1007/s40840-018-0614-0. [8] Y. Z. Dang and J. Sun, Double projection algorithms for solving the split feasibility problems, J. Indus. Manag. Optim., 15 (2019), 2023-2034.  doi: 10.3934/jimo.2018135. [9] Y. Dang, J. Yao and Y. Gao, Relaxed two points projection method for solving the multiple-sets split equality problem, Numer. Algorithms, 78 (2018), 263-275.  doi: 10.1007/s11075-017-0375-0. [10] Y. Z. Dang, J. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Indus. Manag. Optim., 13 (2017), 1383-1394.  doi: 10.3934/jimo.2016078. [11] F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht, 356 (1992), 105–121. [12] A. Gibali, D. T. Mai and N. T. Vinh, A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications, J. Indus. Manag. Optim., 15 (2019), 963-984.  doi: 10.3934/jimo.2018080. [13] B. He, Inexact implicit methods for monotone general variational inequalities, Math. Program., 86 (1999), 199-217.  doi: 10.1007/s101070050086. [14] G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980. [15] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z. [16] B. T. Polyak, Some methods of speeding up the convergence of iterative methods, J. Math. Anal. Appl., 110 (1985), 463-468. [17] B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665.  doi: 10.1088/0266-5611/21/5/009. [18] D. R. Sahu, Y. J. Cho, Q. L. Dong, M. 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. [19] S. Suantai, N. Pholasa and P. Cholamjiak, Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems, Rev. R. Acad. Cienc. Exactas ${F^{ \cdot \cdot\underline a }}$s. Nat. Ser. A Mat. RACSAM, 113 (2019), 1081-1099.  doi: 10.1007/s13398-018-0535-7. [20] G. H. Taddele, P. Kumam, A. G. Gebrie and K. Sitthithakerngkiet, Half-space relaxation projection method for solving multiple-set split feasibility problem, Math. Comput. Appl., 25 (2020), Paper No. 47, 24 pp. doi: 10.3390/mca25030047. [21] W. X. Zhang, D. Han and Z. B. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problems, 25 (2009), 115011, 16 pp. doi: 10.1088/0266-5611/25/11/115001. [22] J. L. Zhao and Q. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Problems, 27 (2011), 035009, 13 pp. doi: 10.1088/0266-5611/27/3/035009.

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References:
 [1] P. K. Anh, D. V. Thong and N. T. Vinh, Improved inertial extragradient methods for solving pseudo-monotone variational inequalities, Optimization, 14 (2020), 1157-1175. [2] N. Buong, P. Hoai and K. T. Binh, Iterative regularization methods for the multiple-sets split feasibility problem in hilbert spaces, Acta Appl. Math., 165 (2020), 183-197.  doi: 10.1007/s10440-019-00249-1. [3] C. Byrne, An unified treatment of some iterative algorithm in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006. [4] 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. [5] Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Math. Program., 42 (1998), 307-325.  doi: 10.1007/BF01589408. [6] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692. [7] P. Cholamjiak, S. Suantai and Su antai S., A new CQ algorithm for solving split feasibility problems in Hilbert spaces, Bull. Malay. Math. Sci. Soc., 42 (2019), 2517-2534.  doi: 10.1007/s40840-018-0614-0. [8] Y. Z. Dang and J. Sun, Double projection algorithms for solving the split feasibility problems, J. Indus. Manag. Optim., 15 (2019), 2023-2034.  doi: 10.3934/jimo.2018135. [9] Y. Dang, J. Yao and Y. Gao, Relaxed two points projection method for solving the multiple-sets split equality problem, Numer. Algorithms, 78 (2018), 263-275.  doi: 10.1007/s11075-017-0375-0. [10] Y. Z. Dang, J. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Indus. Manag. Optim., 13 (2017), 1383-1394.  doi: 10.3934/jimo.2016078. [11] F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht, 356 (1992), 105–121. [12] A. Gibali, D. T. Mai and N. T. Vinh, A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications, J. Indus. Manag. Optim., 15 (2019), 963-984.  doi: 10.3934/jimo.2018080. [13] B. He, Inexact implicit methods for monotone general variational inequalities, Math. Program., 86 (1999), 199-217.  doi: 10.1007/s101070050086. [14] G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980. [15] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z. [16] B. T. Polyak, Some methods of speeding up the convergence of iterative methods, J. Math. Anal. Appl., 110 (1985), 463-468. [17] B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665.  doi: 10.1088/0266-5611/21/5/009. [18] D. R. Sahu, Y. J. Cho, Q. L. Dong, M. 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. [19] S. Suantai, N. Pholasa and P. Cholamjiak, Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems, Rev. R. Acad. Cienc. Exactas ${F^{ \cdot \cdot\underline a }}$s. Nat. Ser. A Mat. RACSAM, 113 (2019), 1081-1099.  doi: 10.1007/s13398-018-0535-7. [20] G. H. Taddele, P. Kumam, A. G. Gebrie and K. Sitthithakerngkiet, Half-space relaxation projection method for solving multiple-set split feasibility problem, Math. Comput. Appl., 25 (2020), Paper No. 47, 24 pp. doi: 10.3390/mca25030047. [21] W. X. Zhang, D. Han and Z. B. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problems, 25 (2009), 115011, 16 pp. doi: 10.1088/0266-5611/25/11/115001. [22] J. L. Zhao and Q. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Problems, 27 (2011), 035009, 13 pp. doi: 10.1088/0266-5611/27/3/035009.
The error results of Example of 4.1 for different initial points
The errors of Example 4.1 with initial point $x^{0} = (0,-1,5)^{T}, t_{k} = 1$ for different inertial factor $\theta_{k}$
The numerical results of Example 4.1
 Initial point CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A})$ DPA with $t_{k} = 1$ Algorithm 3.1 with $t_{k} = 1,\theta_{k} = 0.8$ $x^{0} =\\ (3,2,2)^{T}$ $k = 121; s = 0.0469\\ x^{\ast} = (0.0026;-0.0012;-0.0012)^{T}$ $k = 71; s = 0.3125\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 32; s = 0.\\ x^{\ast} = (1.8244;-0.1457;-0.6787)^{T}$ $x^{0} =\\ (2,3,4)^{T}$ $k = 130; s = 0.0625\\ x^{\ast} = (-0.0035; 0.0001; 0.0036)^{T}$ $k = 73; s = 0.2031\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 10; s = 0.0469\\ x^{\ast} = (-0.80761;0.3253;1.4823)^{T}$ $x^{0} =\\ (0,-1,5)^{T}$ $k = 222; s = 0.0781\\ x^{\ast} = (00031;-0.0054;0.0085)^{T}$ $k = 79; s = 0.1875\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 16; s = 0.0625\\ x^{\ast} = (0.5849;0.2902;0.1249)^{T}$ $x^{0} =\\ (-2,4,3)^{T}$ $k = 219; s = 0.0938\\ x^{\ast} = (-0.0086,0.0055, 0.0031)^{T}$ $k = 73; s = 0.2031\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 24; s = 0.0469\\ x^{\ast} = (-0.0730;0.1899;0.7371)^{T}$
 Initial point CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A})$ DPA with $t_{k} = 1$ Algorithm 3.1 with $t_{k} = 1,\theta_{k} = 0.8$ $x^{0} =\\ (3,2,2)^{T}$ $k = 121; s = 0.0469\\ x^{\ast} = (0.0026;-0.0012;-0.0012)^{T}$ $k = 71; s = 0.3125\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 32; s = 0.\\ x^{\ast} = (1.8244;-0.1457;-0.6787)^{T}$ $x^{0} =\\ (2,3,4)^{T}$ $k = 130; s = 0.0625\\ x^{\ast} = (-0.0035; 0.0001; 0.0036)^{T}$ $k = 73; s = 0.2031\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 10; s = 0.0469\\ x^{\ast} = (-0.80761;0.3253;1.4823)^{T}$ $x^{0} =\\ (0,-1,5)^{T}$ $k = 222; s = 0.0781\\ x^{\ast} = (00031;-0.0054;0.0085)^{T}$ $k = 79; s = 0.1875\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 16; s = 0.0625\\ x^{\ast} = (0.5849;0.2902;0.1249)^{T}$ $x^{0} =\\ (-2,4,3)^{T}$ $k = 219; s = 0.0938\\ x^{\ast} = (-0.0086,0.0055, 0.0031)^{T}$ $k = 73; s = 0.2031\\ x^{\ast} = (0.4281;-0.2507;0.8563)^{T}$ $k = 24; s = 0.0469\\ x^{\ast} = (-0.0730;0.1899;0.7371)^{T}$
The numerical results of Example 4.2
 $M, N$ CQ with stepsize $\frac{1}{\rho(A^{T}A)}$ $t_{k}$ Algorithm3.1 with $\theta_{k} = 0.2$ Algorithm3.1 with $\theta_{k} = 0.8$ DPA $M=20,\\ N=10$ $k=374, s =0.2033$ 0.8 $k=227, s =0.147$ $k=168, s =0.1251$ $k=385, s =0.2203$ 1.0 $k=162, s =0.1052$ $k=120, s =0.0701$ $k=301, s =0.1902$ 1.8 $k=73, s =0.0700$ $k=54, s =0.0512$ $k=284, s =0.1597$ $M = 100,\\ N = 90$ $k = 2695, s = 5.7346$ 0.4 $k = 2431, s = 7.32123$ $k = 2201, s = 6.3075$ $k = 2514, s = 8.4576$ 1 $k = 1071, s = 5.432$ $k = 661, s = 4.1297$ $k = 2303, s = 7.3501$ 1.6 $k = 464, s = 2.6507$ $k = 401, s = 2.1295$ $k = 1998, s = 6.9980$
 $M, N$ CQ with stepsize $\frac{1}{\rho(A^{T}A)}$ $t_{k}$ Algorithm3.1 with $\theta_{k} = 0.2$ Algorithm3.1 with $\theta_{k} = 0.8$ DPA $M=20,\\ N=10$ $k=374, s =0.2033$ 0.8 $k=227, s =0.147$ $k=168, s =0.1251$ $k=385, s =0.2203$ 1.0 $k=162, s =0.1052$ $k=120, s =0.0701$ $k=301, s =0.1902$ 1.8 $k=73, s =0.0700$ $k=54, s =0.0512$ $k=284, s =0.1597$ $M = 100,\\ N = 90$ $k = 2695, s = 5.7346$ 0.4 $k = 2431, s = 7.32123$ $k = 2201, s = 6.3075$ $k = 2514, s = 8.4576$ 1 $k = 1071, s = 5.432$ $k = 661, s = 4.1297$ $k = 2303, s = 7.3501$ 1.6 $k = 464, s = 2.6507$ $k = 401, s = 2.1295$ $k = 1998, s = 6.9980$
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