American Institute of Mathematical Sciences

Advanced
October  2019, 15(4): 2023-2034. doi: 10.3934/jimo.2018135

Double projection algorithms for solving the split feasibility problems

Ya-zheng Dang 1, , Jie Sun 2,3,, and  Su Zhang 2,3,

1. 

School of Management, University of Shanghai for Science and Technology, Shanghai PRC 200093
2. 

Faculty of Science, Curtin University, Benteley, West Australia 6102
3. 

Business School, Nankai University, Tianjin PRC 300071

* Corresponding author: Jie Sun

Received  October 2016 Revised  November 2017 Published  October 2019 Early access  September 2018

Fund Project: This work was partially supported by National Science Foundation of China (Grants 71572113 and 11401322) and Australian Research Council (Grant DP160102819).

We propose two new double projection algorithms for solving the split feasibility problem (SFP). Different from the extragradient projection algorithms, the proposed algorithms do not require fixed stepsize and do not employ the same projection region at different projection steps. We adopt flexible rules for selecting the stepsize and the projection region. The proposed algorithms are shown to be convergent under certain assumptions. Numerical experiments show that the proposed methods appear to be more efficient than the relaxed- CQ algorithm.

Keywords: Armijo-type line search, convergence analysis, double projection algorithm, optimization, split feasibility problem.
Mathematics Subject Classification: Primary: 37C25, 47H09; Secondary: 90C25.
Citation: Ya-zheng Dang, Jie Sun, Su Zhang. Double projection algorithms for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2023-2034. doi: 10.3934/jimo.2018135
References:
[1]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.

[2]

C. Byrne, An unified treatment of some iterative algorithm algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.

[3]

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.

[4]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs, INFORMS Journal on Computing, 16 (2004), 255-265.  doi: 10.1287/ijoc.1030.0046.

[5]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Mathematical Programming, 42 (1998), 307-325.  doi: 10.1007/BF01589408.

[6]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[7]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[8]

L. C. Ceng, Q. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Computers and Mathematics with Applications, 64 (2012), 633-642.  doi: 10.1016/j.camwa.2011.12.074.

[9]

F. Deutsch, The method of alternating orthogonal projections, in Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht, 1992,105-121.

[10]

Y. Dang and Y. Gao, The strong convergence of a KM-CQ-Like algorithm for split feasibility problem, Inverse Problems, 27 (2011), 9pp. doi: 10.1088/0266-5611/27/1/015007.

[11]

Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008.

[12]

B. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 35 (1999), 199-217.  doi: 10.1007/s101070050086.

[13]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.

[15]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.

[16]

B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problem, 21 (2005), 1655-1665.  doi: 10.1088/0266-5611/21/5/009.

[17]

B. Qu and N. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra and Its Application, 428 (2008), 1218-1229.  doi: 10.1016/j.laa.2007.03.002.

[18]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1971.

[19]

H. Xu, A variate Krasnosel$'$ ski-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.

[20]

A. L. Yan, G. Y. Wang and N. Xiu, Robust solutions of split feasibility problem with uncertain linear operator, Journal of Industrial and Management Optimization, 3 (2007), 749-761.  doi: 10.3934/jimo.2007.3.749.

[21]

Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.

[22]

Y. N. Yang, Q. Yang and S. Zhang, Modified alternating direction methods for the modified multiple-sets split feasibility problems, Journal of Optimization Theory and Applications, 163 (2014), 130-147.  doi: 10.1007/s10957-013-0502-6.

[23]

W. X. Zhang, D. Han and Z. B. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problem, 25 (2009), 115001, 16pp. doi: 10.1088/0266-5611/25/11/115001.

[24]

J. L. Zhao and Q. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Problem, 27 (2011), 035009, 13pp. doi: 10.1088/0266-5611/27/3/035009.

[25]

J. Zhao and Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799.  doi: 10.1088/0266-5611/21/5/017.

[26]

E. H. Zarantonello, Projections on convex set in Hilbert space and spectral theory, in Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello), Academic, New York, 1971.

show all references

References:
[1]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.

[2]

C. Byrne, An unified treatment of some iterative algorithm algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.

[3]

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.

[4]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs, INFORMS Journal on Computing, 16 (2004), 255-265.  doi: 10.1287/ijoc.1030.0046.

[5]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Mathematical Programming, 42 (1998), 307-325.  doi: 10.1007/BF01589408.

[6]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[7]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[8]

L. C. Ceng, Q. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Computers and Mathematics with Applications, 64 (2012), 633-642.  doi: 10.1016/j.camwa.2011.12.074.

[9]

F. Deutsch, The method of alternating orthogonal projections, in Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht, 1992,105-121.

[10]

Y. Dang and Y. Gao, The strong convergence of a KM-CQ-Like algorithm for split feasibility problem, Inverse Problems, 27 (2011), 9pp. doi: 10.1088/0266-5611/27/1/015007.

[11]

Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008.

[12]

B. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 35 (1999), 199-217.  doi: 10.1007/s101070050086.

[13]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.

[15]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.

[16]

B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problem, 21 (2005), 1655-1665.  doi: 10.1088/0266-5611/21/5/009.

[17]

B. Qu and N. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra and Its Application, 428 (2008), 1218-1229.  doi: 10.1016/j.laa.2007.03.002.

[18]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1971.

[19]

H. Xu, A variate Krasnosel$'$ ski-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.

[20]

A. L. Yan, G. Y. Wang and N. Xiu, Robust solutions of split feasibility problem with uncertain linear operator, Journal of Industrial and Management Optimization, 3 (2007), 749-761.  doi: 10.3934/jimo.2007.3.749.

[21]

Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.

[22]

Y. N. Yang, Q. Yang and S. Zhang, Modified alternating direction methods for the modified multiple-sets split feasibility problems, Journal of Optimization Theory and Applications, 163 (2014), 130-147.  doi: 10.1007/s10957-013-0502-6.

[23]

W. X. Zhang, D. Han and Z. B. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problem, 25 (2009), 115001, 16pp. doi: 10.1088/0266-5611/25/11/115001.

[24]

J. L. Zhao and Q. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Problem, 27 (2011), 035009, 13pp. doi: 10.1088/0266-5611/27/3/035009.

[25]

J. Zhao and Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799.  doi: 10.1088/0266-5611/21/5/017.

[26]

E. H. Zarantonello, Projections on convex set in Hilbert space and spectral theory, in Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello), Academic, New York, 1971.

Table 1.  The numerical results of Example 5.1
Initiative point CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A})$ Algorithm 3.1 Algorithm 4.1
$ x^{0}=$ $ k=269; s=0.063$ $ k=54; s=0.101$ $ k=28; s=0.077$
$ (-5,-2,-10)^{T}$ $x^{\ast}=(0.5071,-1.8186,01.9072)^{T}$ $x^{\ast}=(0.8718; -1.6577; -1.4080)^{T}$ $x^{\ast}=(1.1595;-1.0082;0-1.0814)^{T}$
$ x^{0}=$ $ k=261; s =0.063$ $ k=16; s=0.061$ $ k=1; s=0.045$
$ (-2,-1,-5)^{T}$ $x^{\ast}=(0.1098;-1.7655; -1.6134)^{T}$ $x^{\ast}=(0.6814;-1.4212; -1.0762)^{T}$ $x^{\ast}=(0.4734;-1.7714 ; -1.3758)^{T}$
$ x^{0}=$ $ k=6450; s =0.525$ $ k=59; s=0.096$ $ k=1; s=0.048$
$(-6,0,-1)^{T}$ $x^{\ast}=(-3.9899;-0.6144; 1.8062)^{T}$ $x^{\ast}=((-3.8898;-0.5850; 1.9604)^{T}$ $x^{\ast}=(-3.9302,-1.0861,1.9786)^{T}$
Table Options

Download as excel

Initiative point CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A})$ Algorithm 3.1 Algorithm 4.1
$ x^{0}=$ $ k=269; s=0.063$ $ k=54; s=0.101$ $ k=28; s=0.077$
$ (-5,-2,-10)^{T}$ $x^{\ast}=(0.5071,-1.8186,01.9072)^{T}$ $x^{\ast}=(0.8718; -1.6577; -1.4080)^{T}$ $x^{\ast}=(1.1595;-1.0082;0-1.0814)^{T}$
$ x^{0}=$ $ k=261; s =0.063$ $ k=16; s=0.061$ $ k=1; s=0.045$
$ (-2,-1,-5)^{T}$ $x^{\ast}=(0.1098;-1.7655; -1.6134)^{T}$ $x^{\ast}=(0.6814;-1.4212; -1.0762)^{T}$ $x^{\ast}=(0.4734;-1.7714 ; -1.3758)^{T}$
$ x^{0}=$ $ k=6450; s =0.525$ $ k=59; s=0.096$ $ k=1; s=0.048$
$(-6,0,-1)^{T}$ $x^{\ast}=(-3.9899;-0.6144; 1.8062)^{T}$ $x^{\ast}=((-3.8898;-0.5850; 1.9604)^{T}$ $x^{\ast}=(-3.9302,-1.0861,1.9786)^{T}$
Table 2.  The numerical results of Example 5.2
$M, N $ CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A}$ $t_{k}$ Algorithm3.1 Algorithm 4.1
$ M=20, N=10$ $ k=485, s =1.040$ 0.8 $ k=274, s =0.312$ $ k=210, s =0.270$
1.0 $ k=193, s =0.100$ $ k=108, s =0.070$
1.8 $ k=103, s =0.067$ $ k=64, s =0.021$
$M=100, N=90$ $ k=3987, s =3.100$ 0.4 $ k=1534, s =0.690$ $ k=1244, s =0.530$
1 $ k=1074, s =0.500$ $ k=630, s =0.261$
1.6 $ k=674, s =0.201$ $ k=412, s =0.132$
Table Options

Download as excel

$M, N $ CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A}$ $t_{k}$ Algorithm3.1 Algorithm 4.1
$ M=20, N=10$ $ k=485, s =1.040$ 0.8 $ k=274, s =0.312$ $ k=210, s =0.270$
1.0 $ k=193, s =0.100$ $ k=108, s =0.070$
1.8 $ k=103, s =0.067$ $ k=64, s =0.021$
$M=100, N=90$ $ k=3987, s =3.100$ 0.4 $ k=1534, s =0.690$ $ k=1244, s =0.530$
1 $ k=1074, s =0.500$ $ k=630, s =0.261$
1.6 $ k=674, s =0.201$ $ k=412, s =0.132$
[1]

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
[2]

Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023
[3]

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
[4]

Ya-Zheng Dang, Zhong-Hui Xue, Yan Gao, Jun-Xiang Li. Fast self-adaptive regularization iterative algorithm for solving split feasibility problem. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1555-1569. doi: 10.3934/jimo.2019017
[5]

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
[6]

Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193
[7]

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
[8]

Suthep Suantai, Nattawut Pholasa, Prasit Cholamjiak. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1595-1615. doi: 10.3934/jimo.2018023
[9]

Zhong Wan, Chaoming Hu, Zhanlu Yang. A spectral PRP conjugate gradient methods for nonconvex optimization problem based on modified line search. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1157-1169. doi: 10.3934/dcdsb.2011.16.1157
[10]

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
[11]

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
[12]

Jianjun Liu, Min Zeng, Yifan Ge, Changzhi Wu, Xiangyu Wang. Improved Cuckoo Search algorithm for numerical function optimization. Journal of Industrial and Management Optimization, 2020, 16 (1) : 103-115. doi: 10.3934/jimo.2018142
[13]

Ayache Benhadid, Fateh Merahi. Complexity analysis of an interior-point algorithm for linear optimization based on a new parametric kernel function with a double barrier term. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022003
[14]

Aviv Gibali, Dang Thi Mai, Nguyen The Vinh. A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. Journal of Industrial and Management Optimization, 2019, 15 (2) : 963-984. doi: 10.3934/jimo.2018080
[15]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
[16]

Mohamed A. Tawhid, Ahmed F. Ali. An effective hybrid firefly algorithm with the cuckoo search for engineering optimization problems. Mathematical Foundations of Computing, 2018, 1 (4) : 349-368. doi: 10.3934/mfc.2018017
[17]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014
[18]

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
[19]

Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial and Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041
[20]

Ouafa Belguidoum, Hassina Grar. An improved projection algorithm for variational inequality problem with multivalued mapping. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022002

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (326)
  • HTML views (1043)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy