# 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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818. [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. [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. [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. [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. [20] R. T. Rockafeller, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. doi: 10.1515/9781400873173. [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. [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. [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. [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. [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. [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. [27] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818. [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. [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. [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. [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. [20] R. T. Rockafeller, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. doi: 10.1515/9781400873173. [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. [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. [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. [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. [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. [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. [27] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.
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
 [1] Guash Haile Taddele, Poom Kumam, Habib ur Rehman, Anteneh Getachew Gebrie. Self adaptive inertial relaxed $CQ$ algorithms for solving split feasibility problem with multiple output sets. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021172 [2] Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092 [3] Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial and Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749 [4] Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022007 [5] Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 [6] Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078 [7] Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187 [8] Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123 [9] Yue Qi, Xiaolin Li, Su Zhang. Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1531-1556. doi: 10.3934/jimo.2020033 [10] Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553 [11] Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5567-5579. doi: 10.3934/dcdsb.2020367 [12] Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021160 [13] Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104 [14] Yazheng Dang, Marcus Ang, Jie Sun. An inertial triple-projection algorithm for solving the split feasibility problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022019 [15] Charles E. M. Pearce, Krzysztof Szajowski, Mitsushi Tamaki. Duration problem with multiple exchanges. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 333-355. doi: 10.3934/naco.2012.2.333 [16] Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311 [17] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 [18] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 [19] Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic and Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029 [20] Weijun Zhou, Youhua Zhou. On the strong convergence of a modified Hestenes-Stiefel method for nonconvex optimization. Journal of Industrial and Management Optimization, 2013, 9 (4) : 893-899. doi: 10.3934/jimo.2013.9.893

2020 Impact Factor: 1.801