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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 |
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[
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. Censor, T. Bortfeld, B. 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. Google Scholar |
[7] | Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008. Google Scholar |
[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. He, C. 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}) $., Soviet Math. Dokl, 269 (1983), 543-547.
|
[13] |
J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, 30, SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719468. |
[14] |
B. Qu and N. Xiu,
A note on the CQ algorithm for the split feasibility problem, Inverse Problem, 21 (2005), 1655-1665.
doi: 10.1088/0266-5611/21/5/009. |
[15] |
B. Recht, M. Fazel and P. Parrilo,
Guaranteed minimum rank solutions of matrix equations via neclear norm minimization, SIAM Rev, 52 (2010), 471-501.
doi: 10.1137/070697835. |
[16] |
A. Sabharwal and L. Potter,
Convexly constrained linear inverse problemsares and regularization, IEEE Trans. Signal Process, 46 (1998), 2345-2352.
doi: 10.1109/78.709518. |
[17] |
H. Xu,
A variate Krasnosel$'$ 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. Censor, T. Bortfeld, B. 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. Google Scholar |
[7] | Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008. Google Scholar |
[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. He, C. 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}) $., Soviet Math. Dokl, 269 (1983), 543-547.
|
[13] |
J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, 30, SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719468. |
[14] |
B. Qu and N. Xiu,
A note on the CQ algorithm for the split feasibility problem, Inverse Problem, 21 (2005), 1655-1665.
doi: 10.1088/0266-5611/21/5/009. |
[15] |
B. Recht, M. Fazel and P. Parrilo,
Guaranteed minimum rank solutions of matrix equations via neclear norm minimization, SIAM Rev, 52 (2010), 471-501.
doi: 10.1137/070697835. |
[16] |
A. Sabharwal and L. Potter,
Convexly constrained linear inverse problemsares and regularization, IEEE Trans. Signal Process, 46 (1998), 2345-2352.
doi: 10.1109/78.709518. |
[17] |
H. Xu,
A variate Krasnosel$'$ 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. |



(N, M) | CQ algorithm | Algorithm 3.1 | Algorithm 4.1 |
(N, M) | CQ algorithm | Algorithm 3.1 | Algorithm 4.1 |
image | Algorithm 3.1 | Algorithm 4.1 |
house | ||
boat | ||
pepper |
image | Algorithm 3.1 | Algorithm 4.1 |
house | ||
boat | ||
pepper |
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