# American Institute of Mathematical Sciences

September  2021, 17(5): 2557-2572. doi: 10.3934/jimo.2020082

## Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem

 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao Shandong, 266590, China 2 School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China

* Corresponding author: Meixia Li

Received  August 2019 Revised  January 2020 Published  September 2021 Early access  April 2020

Fund Project: This project is supported by the Natural Science Foundation of China (Grant No. 11401438, 11571120), Shandong Provincial Natural Science Foundation (Grant No. ZR2017LA002, ZR2019MA022)

The multiple-sets split equality problem is an extended form of the split feasibility problem. It has a wide range of applications in image reconstruction, signal processing, computed tomography, etc. In this paper, we propose a relaxed successive projection algorithm to solve the multiple-sets split equality problem which does not need the prior knowledge of the operator norms, and prove the strong convergence of the algorithm. The numerical examples indicate that the algorithm has good feasibility and effectiveness by comparing with other algorithm.

Citation: Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082
##### References:
 [1] 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.  Google Scholar [2] 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.  Google Scholar [3] S.-S. Chang and R. P. Agarwal, Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 14pp. doi: 10.1186/1029-242X-2014-367.  Google Scholar [4] Y. Censor and T. Elfving, The multiple-sets split feasibility problem and its applicatons for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar [5] Y. Censor, A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244-1256.  doi: 10.1016/j.jmaa.2006.05.010.  Google Scholar [6] S.-S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal., 30 (1997), 4197-4208.  doi: 10.1016/S0362-546X(97)00388-X.  Google Scholar [7] Y.-Z. 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 [8] Y.-Z. Dang, J. Sun and S. Zhang, Double projection algorithms for solving the split feasibility problems, J. Ind. Manag. Optim., 15 (2019), 2023-2034.  doi: 10.3934/jimo.2018135.  Google Scholar [9] Q.-L. Dong and S. He, Self-adaptive projection algorithms for solving the split equality problems, Fixed Point Theory, 18 (2017), 191-202.  doi: 10.24193/fpt-ro.2017.1.15.  Google Scholar [10] Q.-L. Dong, S. He and J. Zhao, Solving the split equality problem without prior knowledge of operator norms, Optimization, 64 (2015), 1887-1906.  doi: 10.1080/02331934.2014.895897.  Google Scholar [11] Y.-Z. 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.  Google Scholar [12] S. Kesornprom, N. Pholasa and P. Cholamjiak, On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem, Numer. Algorithms, 2019 (2019), 1-21.  doi: 10.1007/s11075-019-00790-y.  Google Scholar [13] M. Li, X. Kao and H. Che, Relaxed inertial accelerated algorithms for solving split equality feasibility problem, J. Nonlinear Sci. Appl., 10 (2017), 4109-4121.  doi: 10.22436/jnsa.010.08.07.  Google Scholar [14] A. Moudafi and A. Gibali, $l_1$-$l_2$ regularization of split feasibility problems, Numer. Algorithms, 78 (2018), 739-757.  doi: 10.1007/s11075-017-0398-6.  Google Scholar [15] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818.   Google Scholar [16] 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 [17] B. Qu, C. Wang and N. Xiu, Analysis on Newton projection method for the split feasibility problem, Comput. Optim. Appl., 67 (2017), 175-199.  doi: 10.1007/s10589-016-9884-3.  Google Scholar [18] B. Qu, B. Liu and N. Zheng, On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218-223.  doi: 10.1016/j.amc.2015.04.056.  Google Scholar [19] B. Qu and H. Chang, Remark on the successive projection algorithm for the multiple-sets split feasibility problem, Numer. Funct. Anal. Optim., 38 (2017), 1614-1623.  doi: 10.1080/01630563.2017.1369109.  Google Scholar [20] R. T. Rockafeller, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. doi: 10.1515/9781400873173.  Google Scholar [21] S. Suantai, S. Kesornprom and P. Cholamjiak, A new hybrid CQ algorithm for the split feasibility problem in Hilbert spaces and its applications to compressed sensing, Math., 7 (2019), 15pp. doi: 10.3390/math7090789.  Google Scholar [22] S. Suantai, N. 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.  Google Scholar [23] 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 Fís. Nat. Ser. A Mat. RACSAM, 113 (2019), 1081-1099.  doi: 10.1007/s13398-018-0535-7.  Google Scholar [24] L. Shi, R. Chen and Y. Wu, An iterative algorithm for the split equality and multiple-sets split equality problem, Abstr. Appl. Anal., 2014 (2014), 5pp. doi: 10.1155/2014/620813.  Google Scholar [25] N. T. Vinh, P. Cholamjiak and S. Suantai, A new CQ algorithm for solving split feasibility problems in Hilbert spaces, Bull. Malays. Math. Sci. Soc., 42 (2019), 2517-2534.  doi: 10.1007/s40840-018-0614-0.  Google Scholar [26] Y. Wu, R. Chen and L. Shi, Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings, J. Inequal. Appl., 2014 (2014), 8pp. doi: 10.1186/1029-242X-2014-428.  Google Scholar [27] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar

show all references

##### References:
 [1] 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.  Google Scholar [2] 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.  Google Scholar [3] S.-S. Chang and R. P. Agarwal, Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 14pp. doi: 10.1186/1029-242X-2014-367.  Google Scholar [4] Y. Censor and T. Elfving, The multiple-sets split feasibility problem and its applicatons for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar [5] Y. Censor, A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244-1256.  doi: 10.1016/j.jmaa.2006.05.010.  Google Scholar [6] S.-S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal., 30 (1997), 4197-4208.  doi: 10.1016/S0362-546X(97)00388-X.  Google Scholar [7] Y.-Z. 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 [8] Y.-Z. Dang, J. Sun and S. Zhang, Double projection algorithms for solving the split feasibility problems, J. Ind. Manag. Optim., 15 (2019), 2023-2034.  doi: 10.3934/jimo.2018135.  Google Scholar [9] Q.-L. Dong and S. He, Self-adaptive projection algorithms for solving the split equality problems, Fixed Point Theory, 18 (2017), 191-202.  doi: 10.24193/fpt-ro.2017.1.15.  Google Scholar [10] Q.-L. Dong, S. He and J. Zhao, Solving the split equality problem without prior knowledge of operator norms, Optimization, 64 (2015), 1887-1906.  doi: 10.1080/02331934.2014.895897.  Google Scholar [11] Y.-Z. 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.  Google Scholar [12] S. Kesornprom, N. Pholasa and P. Cholamjiak, On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem, Numer. Algorithms, 2019 (2019), 1-21.  doi: 10.1007/s11075-019-00790-y.  Google Scholar [13] M. Li, X. Kao and H. Che, Relaxed inertial accelerated algorithms for solving split equality feasibility problem, J. Nonlinear Sci. Appl., 10 (2017), 4109-4121.  doi: 10.22436/jnsa.010.08.07.  Google Scholar [14] A. Moudafi and A. Gibali, $l_1$-$l_2$ regularization of split feasibility problems, Numer. Algorithms, 78 (2018), 739-757.  doi: 10.1007/s11075-017-0398-6.  Google Scholar [15] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818.   Google Scholar [16] 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 [17] B. Qu, C. Wang and N. Xiu, Analysis on Newton projection method for the split feasibility problem, Comput. Optim. Appl., 67 (2017), 175-199.  doi: 10.1007/s10589-016-9884-3.  Google Scholar [18] B. Qu, B. Liu and N. Zheng, On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218-223.  doi: 10.1016/j.amc.2015.04.056.  Google Scholar [19] B. Qu and H. Chang, Remark on the successive projection algorithm for the multiple-sets split feasibility problem, Numer. Funct. Anal. Optim., 38 (2017), 1614-1623.  doi: 10.1080/01630563.2017.1369109.  Google Scholar [20] R. T. Rockafeller, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. doi: 10.1515/9781400873173.  Google Scholar [21] S. Suantai, S. Kesornprom and P. Cholamjiak, A new hybrid CQ algorithm for the split feasibility problem in Hilbert spaces and its applications to compressed sensing, Math., 7 (2019), 15pp. doi: 10.3390/math7090789.  Google Scholar [22] S. Suantai, N. 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.  Google Scholar [23] 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 Fís. Nat. Ser. A Mat. RACSAM, 113 (2019), 1081-1099.  doi: 10.1007/s13398-018-0535-7.  Google Scholar [24] L. Shi, R. Chen and Y. Wu, An iterative algorithm for the split equality and multiple-sets split equality problem, Abstr. Appl. Anal., 2014 (2014), 5pp. doi: 10.1155/2014/620813.  Google Scholar [25] N. T. Vinh, P. Cholamjiak and S. Suantai, A new CQ algorithm for solving split feasibility problems in Hilbert spaces, Bull. Malays. Math. Sci. Soc., 42 (2019), 2517-2534.  doi: 10.1007/s40840-018-0614-0.  Google Scholar [26] Y. Wu, R. Chen and L. Shi, Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings, J. Inequal. Appl., 2014 (2014), 8pp. doi: 10.1186/1029-242X-2014-428.  Google Scholar [27] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar
The iteration number of RSPA and RTPP in Case A for Example 4.1
The iteration number of RSPA and RTPP in Case B for Example 4.1
The iteration number of RSPA and RTPP in Case C for Example 4.2
The iteration number of RSPA and RTPP in Case D for Example 4.2
The numerical results of Example 4.1
 Init. $x_1=(1,1,1)^T$ $y_1=(1,1,1,1)^T$ $n=14, s=0.003312$ RSPA $x^*=(0.744,1.802,-0.223)^T*10^{-5}$ $y^*=(-1.770,7.964,-0.050,-1.210)^T*10^{-5}$ $n=327, s=0.009660$ RTPP $x^*=(0.280,-0.166,0.319)^T$ $y^*=(-0.190,0.477,0.336,0.207)^T$
 Init. $x_1=(1,1,1)^T$ $y_1=(1,1,1,1)^T$ $n=14, s=0.003312$ RSPA $x^*=(0.744,1.802,-0.223)^T*10^{-5}$ $y^*=(-1.770,7.964,-0.050,-1.210)^T*10^{-5}$ $n=327, s=0.009660$ RTPP $x^*=(0.280,-0.166,0.319)^T$ $y^*=(-0.190,0.477,0.336,0.207)^T$
The numerical results of Example 4.1
 Init. $x_1=10(1,1,1)^T$ $y_1=10(1,1,1,1)^T$ $n=30, s=0.005835$ RSPA $x^*=(0.285,2.783,-0.0856)^T*10^{-5}$ $y^*=(-2.730,-2.635,-1.595,7.311)^T*10^{-5}$ $n=54878, s=1.100722$ RTPP $x^*=(6.751;-10.660;10.159)^T$ $y^*=(3.244;6.605;5.986;1.070)^T$
 Init. $x_1=10(1,1,1)^T$ $y_1=10(1,1,1,1)^T$ $n=30, s=0.005835$ RSPA $x^*=(0.285,2.783,-0.0856)^T*10^{-5}$ $y^*=(-2.730,-2.635,-1.595,7.311)^T*10^{-5}$ $n=54878, s=1.100722$ RTPP $x^*=(6.751;-10.660;10.159)^T$ $y^*=(3.244;6.605;5.986;1.070)^T$
The numerical results of Example 4.1
 Init. $x_1=-10(1,1,1)^T$ $y_1=10(1,1,1,1)^T$ $n=29, s=0.005965$ RSPA $x^*=(0.740,4.800,-0.222)^T*10^{-5}$ $y^*=(-0.482,-0.456,-0.289,1.280)^T*10^{-4}$ $n=907710, s=1.785933$ RTPP $x^*=(1.128,-1.722,0.520)^T$ $y^*=(0.096,1.365,3.149,-2.396)^T$
 Init. $x_1=-10(1,1,1)^T$ $y_1=10(1,1,1,1)^T$ $n=29, s=0.005965$ RSPA $x^*=(0.740,4.800,-0.222)^T*10^{-5}$ $y^*=(-0.482,-0.456,-0.289,1.280)^T*10^{-4}$ $n=907710, s=1.785933$ RTPP $x^*=(1.128,-1.722,0.520)^T$ $y^*=(0.096,1.365,3.149,-2.396)^T$
The numerical results of Example 4.2
 RSPA RTPP $J$ $N$ $M$ $n$ $s$ $n$ $s$ $10$ $20$ $30$ $13$ $0.001976$ 1370 0.078588 Case 1 40 30 40 14 0.002048 20842 3.989155 60 60 60 15 0.002840 24600 10.765349 $10$ $20$ $30$ 15 0.001523 9573 0.669758 Case 2 40 30 40 17 0.002967 21674 4.326832 60 60 60 18 0.003256 23970 12.725284 $10$ $20$ $30$ 16 0.001644 1338 0.078992 Case 3 40 30 40 17 0.001897 21237 4.291747 60 60 60 18 0.003552 24110 10.261271 $10$ $20$ $30$ 15 0.001891 9573 0.528336 Case 4 40 30 40 17 0.002379 21674 4.199953 60 60 60 18 0.002865 23970 10.365368
 RSPA RTPP $J$ $N$ $M$ $n$ $s$ $n$ $s$ $10$ $20$ $30$ $13$ $0.001976$ 1370 0.078588 Case 1 40 30 40 14 0.002048 20842 3.989155 60 60 60 15 0.002840 24600 10.765349 $10$ $20$ $30$ 15 0.001523 9573 0.669758 Case 2 40 30 40 17 0.002967 21674 4.326832 60 60 60 18 0.003256 23970 12.725284 $10$ $20$ $30$ 16 0.001644 1338 0.078992 Case 3 40 30 40 17 0.001897 21237 4.291747 60 60 60 18 0.003552 24110 10.261271 $10$ $20$ $30$ 15 0.001891 9573 0.528336 Case 4 40 30 40 17 0.002379 21674 4.199953 60 60 60 18 0.002865 23970 10.365368
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