# American Institute of Mathematical Sciences

• Previous Article
Stochastic-Lazier-Greedy Algorithm for monotone non-submodular maximization
• JIMO Home
• This Issue
• Next Article
Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition
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  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, 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
 [1] Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367 [2] Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 [3] Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391 [4] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [5] Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122 [6] Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 [7] Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 [8] Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353 [9] Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170 [10] Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker. Chaotic Delone sets. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021016 [11] Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 [12] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [13] Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285 [14] José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 [15] Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015 [16] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [17] Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 [18] Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021002 [19] Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314 [20] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

2019 Impact Factor: 1.366

## Tools

Article outline

Figures and Tables