# American Institute of Mathematical Sciences

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

## Double projection algorithms for solving the split feasibility problems

 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  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.

Citation: Ya-zheng Dang, Jie Sun, Su Zhang. Double projection algorithms for solving the split feasibility problems. Journal of Industrial & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] F. Deutsch, The method of alternating orthogonal projections, in Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht, 1992,105-121. Google Scholar [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. Google Scholar [11] Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008.Google Scholar [12] B. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 35 (1999), 199-217. doi: 10.1007/s101070050086. Google Scholar [13] G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980. Google Scholar [14] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1971. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] F. Deutsch, The method of alternating orthogonal projections, in Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht, 1992,105-121. Google Scholar [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. Google Scholar [11] Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008.Google Scholar [12] B. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 35 (1999), 199-217. doi: 10.1007/s101070050086. Google Scholar [13] G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980. Google Scholar [14] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1971. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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 $x^{0}=$ $k=269; s=0.063$ $k=54; s=0.101$ $k=28; s=0.077$ $(-5,-2,-10)^{T}$ $x^{\ast}=(0.5071,-1.8186,01.9072)^{T}$ $x^{\ast}=(0.8718; -1.6577; -1.4080)^{T}$ $x^{\ast}=(1.1595;-1.0082;0-1.0814)^{T}$ $x^{0}=$ $k=261; s =0.063$ $k=16; s=0.061$ $k=1; s=0.045$ $(-2,-1,-5)^{T}$ $x^{\ast}=(0.1098;-1.7655; -1.6134)^{T}$ $x^{\ast}=(0.6814;-1.4212; -1.0762)^{T}$ $x^{\ast}=(0.4734;-1.7714 ; -1.3758)^{T}$ $x^{0}=$ $k=6450; s =0.525$ $k=59; s=0.096$ $k=1; s=0.048$ $(-6,0,-1)^{T}$ $x^{\ast}=(-3.9899;-0.6144; 1.8062)^{T}$ $x^{\ast}=((-3.8898;-0.5850; 1.9604)^{T}$ $x^{\ast}=(-3.9302,-1.0861,1.9786)^{T}$
 Initiative point CQ Algorithm with stepsize $\frac{1}{\rho(A^{T}A})$ Algorithm 3.1 Algorithm 4.1 $x^{0}=$ $k=269; s=0.063$ $k=54; s=0.101$ $k=28; s=0.077$ $(-5,-2,-10)^{T}$ $x^{\ast}=(0.5071,-1.8186,01.9072)^{T}$ $x^{\ast}=(0.8718; -1.6577; -1.4080)^{T}$ $x^{\ast}=(1.1595;-1.0082;0-1.0814)^{T}$ $x^{0}=$ $k=261; s =0.063$ $k=16; s=0.061$ $k=1; s=0.045$ $(-2,-1,-5)^{T}$ $x^{\ast}=(0.1098;-1.7655; -1.6134)^{T}$ $x^{\ast}=(0.6814;-1.4212; -1.0762)^{T}$ $x^{\ast}=(0.4734;-1.7714 ; -1.3758)^{T}$ $x^{0}=$ $k=6450; s =0.525$ $k=59; s=0.096$ $k=1; s=0.048$ $(-6,0,-1)^{T}$ $x^{\ast}=(-3.9899;-0.6144; 1.8062)^{T}$ $x^{\ast}=((-3.8898;-0.5850; 1.9604)^{T}$ $x^{\ast}=(-3.9302,-1.0861,1.9786)^{T}$
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 $k=674, s =0.201$ $k=412, s =0.132$
 $M, N$ 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 $k=674, s =0.201$ $k=412, s =0.132$
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