July  2020, 16(4): 1555-1569. doi: 10.3934/jimo.2019017

Fast self-adaptive regularization iterative algorithm for solving split feasibility problem

1. 

School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

Shanghai Publication and Printing College, Shanghai 200093, China

* Corresponding author

Received  June 2018 Revised  September 2018 Published  July 2020 Early access  March 2019

Split feasibility problem (SFP) is to find a point which belongs to one convex set in one space, such that its image under a linear transformation belongs to another convex set in the image space. This paper deals with a unified regularized SFP for the convex case. We first construct a self-adaptive regularization iterative algorithm by using Armijo-like search for the SFP and show it converges at a subliner rate of $ O(1/k) $, where $ k $ represents the number of iterations. More interestingly, inspired by the acceleration technique introduced by Nesterov[12], we present a fast Armijo-like regularization iterative algorithm and show it converges at rate of $ O(1/k^{2}) $. The efficiency of the algorithms is demonstrated by some random data and image debluring problems.

Citation: 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
References:
[1]

C. Byrne, A 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.

[2]

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.

[3]

Y. CensorT. BortfeldB. Mortin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation, Physics in Medicine and Biology, 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.

[4]

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.

[5]

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

[6]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Spring Netherlands, 2000.

[7] Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008. 
[8]

P.C. Hansen, J. G. Nagy and D. P. Oleary, Deblurring Images: Matices, Spectra and Giltering, SIAM, 2006. doi: 10.1137/1.9780898718874.

[9]

H. J. HeC. Ling and H. Xu, An implementable splitting algorithm for the $ l_{1} $ norm regularized split feasibility problem, Journal of Scientific Computing, 67 (2016), 281-298.  doi: 10.1007/s10915-015-0078-4.

[10]

S. N. He and W. L. Zhu, A note on approximating curve with 1-norm regularization method for the split feasibility problem, Journal of Applied Mathematics, 2012 (2012), Article ID 683890, 10 pages. doi: 10.1155/2012/683890.

[11]

M. Li, Improved relaxed CQ methods for solving the split feasibility problem, Advanced Modeling and Optimization, 13 (2011), 305-317. 

[12]

Y. Nesterov, A method for solving a convex programming problems with convergence rate $ O(1/k^{2}) $., Soviet Math. Dokl, 269 (1983), 543-547. 

[13]

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, 30, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719468.

[14]

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.

[15]

B. RechtM. Fazel and P. Parrilo, Guaranteed minimum rank solutions of matrix equations via neclear norm minimization, SIAM Rev, 52 (2010), 471-501.  doi: 10.1137/070697835.

[16]

A. Sabharwal and L. Potter, Convexly constrained linear inverse problemsares and regularization, IEEE Trans. Signal Process, 46 (1998), 2345-2352.  doi: 10.1109/78.709518.

[17]

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.

[18]

H. Xu and F. Wang, Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem, Journal of Inequalities and Applicatio, 2010 (2010), Article ID102085, 13 pages. doi: 10.1155/2010/102085.

[19]

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.

[20]

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.

show all references

References:
[1]

C. Byrne, A 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.

[2]

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.

[3]

Y. CensorT. BortfeldB. Mortin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation, Physics in Medicine and Biology, 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.

[4]

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.

[5]

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

[6]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Spring Netherlands, 2000.

[7] Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008. 
[8]

P.C. Hansen, J. G. Nagy and D. P. Oleary, Deblurring Images: Matices, Spectra and Giltering, SIAM, 2006. doi: 10.1137/1.9780898718874.

[9]

H. J. HeC. Ling and H. Xu, An implementable splitting algorithm for the $ l_{1} $ norm regularized split feasibility problem, Journal of Scientific Computing, 67 (2016), 281-298.  doi: 10.1007/s10915-015-0078-4.

[10]

S. N. He and W. L. Zhu, A note on approximating curve with 1-norm regularization method for the split feasibility problem, Journal of Applied Mathematics, 2012 (2012), Article ID 683890, 10 pages. doi: 10.1155/2012/683890.

[11]

M. Li, Improved relaxed CQ methods for solving the split feasibility problem, Advanced Modeling and Optimization, 13 (2011), 305-317. 

[12]

Y. Nesterov, A method for solving a convex programming problems with convergence rate $ O(1/k^{2}) $., Soviet Math. Dokl, 269 (1983), 543-547. 

[13]

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, 30, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719468.

[14]

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.

[15]

B. RechtM. Fazel and P. Parrilo, Guaranteed minimum rank solutions of matrix equations via neclear norm minimization, SIAM Rev, 52 (2010), 471-501.  doi: 10.1137/070697835.

[16]

A. Sabharwal and L. Potter, Convexly constrained linear inverse problemsares and regularization, IEEE Trans. Signal Process, 46 (1998), 2345-2352.  doi: 10.1109/78.709518.

[17]

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.

[18]

H. Xu and F. Wang, Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem, Journal of Inequalities and Applicatio, 2010 (2010), Article ID102085, 13 pages. doi: 10.1155/2010/102085.

[19]

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.

[20]

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.

Figure 1.  Evolutions of SNR with respect to iterations
Figure 2.  Evolutions of objective value with respect to iterations
Figure 3.  In each subgraph (a or b or c) top(from left to right): clean images and corrupted images, respectively; bottom(from left to right): Recovered images by Algorithm 3.1 and Algorithm 4.1, respectively. house.png (256 × 256); boat.png (256 × 256); pepper.png (256 × 256).
Table 1.  Numerical comparison for synthetic date
(N, M) CQ algorithm Algorithm 3.1 Algorithm 4.1
$ (10, 20) $ $ Iter.=13673; s =0.6915 $ $ Iter.=5019; s=0.5747 $ $ Iter.=565; s=0.0553 $
$ (20, 40) $ $ Iter.=23694; s =0.9182 $ $ Iter.=2752; s =0.3956 $ $ Iter.=1553; s=0.3027 $
$ (50, 50) $ $ Iter.=49389; s =1.9104 $ $ Iter.=16794, s =1.1770 $ $ Iter.=1407; s=0.4170 $
(N, M) CQ algorithm Algorithm 3.1 Algorithm 4.1
$ (10, 20) $ $ Iter.=13673; s =0.6915 $ $ Iter.=5019; s=0.5747 $ $ Iter.=565; s=0.0553 $
$ (20, 40) $ $ Iter.=23694; s =0.9182 $ $ Iter.=2752; s =0.3956 $ $ Iter.=1553; s=0.3027 $
$ (50, 50) $ $ Iter.=49389; s =1.9104 $ $ Iter.=16794, s =1.1770 $ $ Iter.=1407; s=0.4170 $
Table 2.  The numerical results for image deblurring
image Algorithm 3.1 Algorithm 4.1
house $ Iter.=645, s =35.6395 $ $ SNR=24.0820 $ $ Iter.=357, s =13.8825 $ $ SNR=24.2712 $
boat $ Iter.=800, s =133.7814 $ $ SNR=21.2978 $ $ Iter.=530, s =42.8210 $$ SNR= 21.4350 $
pepper $ Iter.=1026, s =57.6502 $ $ SNR=20.1239 $ $ Iter.=588, s =30.8807 $ $ SNR= 20.2807 $
image Algorithm 3.1 Algorithm 4.1
house $ Iter.=645, s =35.6395 $ $ SNR=24.0820 $ $ Iter.=357, s =13.8825 $ $ SNR=24.2712 $
boat $ Iter.=800, s =133.7814 $ $ SNR=21.2978 $ $ Iter.=530, s =42.8210 $$ SNR= 21.4350 $
pepper $ Iter.=1026, s =57.6502 $ $ SNR=20.1239 $ $ Iter.=588, s =30.8807 $ $ SNR= 20.2807 $
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