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A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications
1. | Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel |
2. | The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838 Haifa, Israel |
3. | Department of Mathematics, University of Transport and Communications, 3 Cau Giay Street, Hanoi, Vietnam |
Inspired by the works of López et al. [
References:
[1] |
F. Alvarez and H. Attouch,
An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.
doi: 10.1023/A:1011253113155. |
[2] |
Q. H. Ansari and A. Rehan, Split feasibility and fixed point problems, In: Q. H. Ansari (ed.),
Nonlinear Anal. Approx. The., Optim. Appl., Springer, (2014), 281-322. |
[3] |
J. P. Aubin,
Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993.
doi: 10.1007/978-3-662-02959-6. |
[4] |
K. J. Batenburg,
A network flow algorithm for reconstructing binary images from discrete X-rays, J. Math. Imaging Vis., 27 (2007), 175-191.
doi: 10.1007/s10851-006-9798-2. |
[5] |
D. Butnariu and A. N. Iusem,
Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
doi: 10.1007/978-94-011-4066-9. |
[6] |
H. H. Bauschke and J. M. Borwein,
On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.
doi: 10.1137/S0036144593251710. |
[7] |
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. |
[8] |
C. Byrne,
A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.
doi: 10.1088/0266-5611/20/1/006. |
[9] |
L. C. Ceng, Q. H. Ansari and J. C. Yao,
An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642.
doi: 10.1016/j.camwa.2011.12.074. |
[10] |
Y. Censor and T. Elfving,
A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.
doi: 10.1007/BF02142692. |
[11] |
Y. Censor, T. Bortfeld, B. Martin and A. Trofimov,
A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2003), 2353-2365.
doi: 10.1088/0031-9155/51/10/001. |
[12] |
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. |
[13] |
S. Chen, D. Donoho and M. Saunders,
Atomic decomposition by basis pursuit, SIAM J. Comput., 20 (1998), 33-61.
doi: 10.1137/S1064827596304010. |
[14] |
Y. Dang and Y. Gao,
The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (2011), 015007, 9 pp.
doi: 10.1088/0266-5611/27/1/015007. |
[15] |
Y. Dang, J. Sun and H. K. Xu,
Inertial accelerated algorithms for solving a split feasibility problem, J. Indus. Manage. Optim., 13 (2017), 1383-1394.
doi: 10.3934/jimo.2016078. |
[16] |
A. Gibali, L.-W. Liu and Y.-C. Tang,
Note on the modified relaxation CQ algorithm for the split feasibility problem, Optim. Lett., (2017), 1-14.
doi: 10.1007/s11590-017-1148-3. |
[17] |
A. Gibali and S. Petra, DC-Programming versus l0-Superiorization for Discrete Tomography, To appear in Analele Ştiinţifice ale Universitatii Ovidius Constanţa, Seria Mathematica, 2017. |
[18] |
K. Goebel and S. Reich,
Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984. |
[19] |
P. C. Hansen and M. Saxild-Hansen,
AIR tools - a MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math., 236 (2012), 2167-2178.
doi: 10.1016/j.cam.2011.09.039. |
[20] |
M. Li and Y. Yao,
Strong convergence of an iterative algorithm for λ-strictly pseudocontractive mappings in Hilbert spaces, Analele Ştiinţifice ale Universitatii Ovidius Constanţa, Seria Mathematica, 18 (2010), 219-228.
|
[21] |
G. López, V. Martín-Márquez, F. Wang and H. K. Xu,
Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Problems, 28 (2012), 085004, 18 pp.
doi: 10.1088/0266-5611/28/8/085004. |
[22] |
Y. Lou and M. Yan, Fast l1 - l2 Minimization via a proximal operator, arXiv: 1609.09530. |
[23] |
P. E. Maingé,
Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.
doi: 10.1016/j.jmaa.2005.12.066. |
[24] |
P. E. Maingé,
Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.
doi: 10.1016/j.cam.2007.07.021. |
[25] |
P. E. Maingé,
Strong convergence of projected subgradientmethods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.
doi: 10.1007/s11228-008-0102-z. |
[26] |
A. Moudafi and A. Gibali,
l1-l2 Regularization of split feasibility problems, Numer. Algorithms, (2017), 1-19.
doi: 10.1007/s11075-017-0398-6. |
[27] |
T. L. N. Nguyen and Y. Shin, Deterministic sensing matrices in compressive sensing: A
survey, Sci. World J., 2013 (2013), Article ID 192795, 6 pages.
doi: 10.1155/2013/192795. |
[28] |
Z. Opial,
Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc., 73 (1967), 591-597.
doi: 10.1090/S0002-9904-1967-11761-0. |
[29] |
Y. Shehu and O. S. Iyiola,
Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems, Optimization, 67 (2018), 475-492.
doi: 10.1080/02331934.2017.1405955. |
[30] |
Y. Shehu and O. S. Iyiola,
Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510.
doi: 10.1007/s11784-017-0435-z. |
[31] |
S. Suantai, N. Pholasa and P. Cholamjiak,
The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., (2017).
doi: 10.3934/jimo.2018023. |
[32] |
R. Tibshirani,
Regression shrinkage and selection Via the lasso, J. Royal Stat. Soc., 58 (1996), 267-288.
|
[33] |
F. Wang and H.-K. Xu,
Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105-4111.
doi: 10.1016/j.na.2011.03.044. |
[34] |
F. Wang,
Polyak's gradient method for split feasibility problem constrained by level sets, Numer. Algorithms, 77 (2018), 925-938.
doi: 10.1007/s11075-017-0347-4. |
[35] |
H.-K. Xu,
Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 105018, 17 pp.
doi: 10.1088/0266-5611/26/10/105018. |
[36] |
H.-K. Xu,
Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.
doi: 10.1112/S0024610702003332. |
[37] |
Z. Xu, X. Chang, F. Xu and H. Zhang,
l1-2 Regularization: A thresholding representation theory and a fast solver, IEEE Trans. Neural Netw. Learn. Syst., 23 (2012), 1013-1027.
|
[38] |
Q. Yang,
The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.
doi: 10.1088/0266-5611/20/4/014. |
[39] |
H. Zhou and P. Wang,
Adaptively relaxed algorithms for solving the split feasibility problem with a new step size, J. Inequal. Appl., 2014 (20214), 22pp.
doi: 10.1186/1029-242X-2014-448. |
show all references
References:
[1] |
F. Alvarez and H. Attouch,
An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.
doi: 10.1023/A:1011253113155. |
[2] |
Q. H. Ansari and A. Rehan, Split feasibility and fixed point problems, In: Q. H. Ansari (ed.),
Nonlinear Anal. Approx. The., Optim. Appl., Springer, (2014), 281-322. |
[3] |
J. P. Aubin,
Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993.
doi: 10.1007/978-3-662-02959-6. |
[4] |
K. J. Batenburg,
A network flow algorithm for reconstructing binary images from discrete X-rays, J. Math. Imaging Vis., 27 (2007), 175-191.
doi: 10.1007/s10851-006-9798-2. |
[5] |
D. Butnariu and A. N. Iusem,
Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
doi: 10.1007/978-94-011-4066-9. |
[6] |
H. H. Bauschke and J. M. Borwein,
On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.
doi: 10.1137/S0036144593251710. |
[7] |
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. |
[8] |
C. Byrne,
A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.
doi: 10.1088/0266-5611/20/1/006. |
[9] |
L. C. Ceng, Q. H. Ansari and J. C. Yao,
An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642.
doi: 10.1016/j.camwa.2011.12.074. |
[10] |
Y. Censor and T. Elfving,
A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.
doi: 10.1007/BF02142692. |
[11] |
Y. Censor, T. Bortfeld, B. Martin and A. Trofimov,
A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2003), 2353-2365.
doi: 10.1088/0031-9155/51/10/001. |
[12] |
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. |
[13] |
S. Chen, D. Donoho and M. Saunders,
Atomic decomposition by basis pursuit, SIAM J. Comput., 20 (1998), 33-61.
doi: 10.1137/S1064827596304010. |
[14] |
Y. Dang and Y. Gao,
The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (2011), 015007, 9 pp.
doi: 10.1088/0266-5611/27/1/015007. |
[15] |
Y. Dang, J. Sun and H. K. Xu,
Inertial accelerated algorithms for solving a split feasibility problem, J. Indus. Manage. Optim., 13 (2017), 1383-1394.
doi: 10.3934/jimo.2016078. |
[16] |
A. Gibali, L.-W. Liu and Y.-C. Tang,
Note on the modified relaxation CQ algorithm for the split feasibility problem, Optim. Lett., (2017), 1-14.
doi: 10.1007/s11590-017-1148-3. |
[17] |
A. Gibali and S. Petra, DC-Programming versus l0-Superiorization for Discrete Tomography, To appear in Analele Ştiinţifice ale Universitatii Ovidius Constanţa, Seria Mathematica, 2017. |
[18] |
K. Goebel and S. Reich,
Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984. |
[19] |
P. C. Hansen and M. Saxild-Hansen,
AIR tools - a MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math., 236 (2012), 2167-2178.
doi: 10.1016/j.cam.2011.09.039. |
[20] |
M. Li and Y. Yao,
Strong convergence of an iterative algorithm for λ-strictly pseudocontractive mappings in Hilbert spaces, Analele Ştiinţifice ale Universitatii Ovidius Constanţa, Seria Mathematica, 18 (2010), 219-228.
|
[21] |
G. López, V. Martín-Márquez, F. Wang and H. K. Xu,
Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Problems, 28 (2012), 085004, 18 pp.
doi: 10.1088/0266-5611/28/8/085004. |
[22] |
Y. Lou and M. Yan, Fast l1 - l2 Minimization via a proximal operator, arXiv: 1609.09530. |
[23] |
P. E. Maingé,
Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.
doi: 10.1016/j.jmaa.2005.12.066. |
[24] |
P. E. Maingé,
Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.
doi: 10.1016/j.cam.2007.07.021. |
[25] |
P. E. Maingé,
Strong convergence of projected subgradientmethods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.
doi: 10.1007/s11228-008-0102-z. |
[26] |
A. Moudafi and A. Gibali,
l1-l2 Regularization of split feasibility problems, Numer. Algorithms, (2017), 1-19.
doi: 10.1007/s11075-017-0398-6. |
[27] |
T. L. N. Nguyen and Y. Shin, Deterministic sensing matrices in compressive sensing: A
survey, Sci. World J., 2013 (2013), Article ID 192795, 6 pages.
doi: 10.1155/2013/192795. |
[28] |
Z. Opial,
Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc., 73 (1967), 591-597.
doi: 10.1090/S0002-9904-1967-11761-0. |
[29] |
Y. Shehu and O. S. Iyiola,
Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems, Optimization, 67 (2018), 475-492.
doi: 10.1080/02331934.2017.1405955. |
[30] |
Y. Shehu and O. S. Iyiola,
Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510.
doi: 10.1007/s11784-017-0435-z. |
[31] |
S. Suantai, N. Pholasa and P. Cholamjiak,
The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., (2017).
doi: 10.3934/jimo.2018023. |
[32] |
R. Tibshirani,
Regression shrinkage and selection Via the lasso, J. Royal Stat. Soc., 58 (1996), 267-288.
|
[33] |
F. Wang and H.-K. Xu,
Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105-4111.
doi: 10.1016/j.na.2011.03.044. |
[34] |
F. Wang,
Polyak's gradient method for split feasibility problem constrained by level sets, Numer. Algorithms, 77 (2018), 925-938.
doi: 10.1007/s11075-017-0347-4. |
[35] |
H.-K. Xu,
Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 105018, 17 pp.
doi: 10.1088/0266-5611/26/10/105018. |
[36] |
H.-K. Xu,
Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.
doi: 10.1112/S0024610702003332. |
[37] |
Z. Xu, X. Chang, F. Xu and H. Zhang,
l1-2 Regularization: A thresholding representation theory and a fast solver, IEEE Trans. Neural Netw. Learn. Syst., 23 (2012), 1013-1027.
|
[38] |
Q. Yang,
The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.
doi: 10.1088/0266-5611/20/4/014. |
[39] |
H. Zhou and P. Wang,
Adaptively relaxed algorithms for solving the split feasibility problem with a new step size, J. Inequal. Appl., 2014 (20214), 22pp.
doi: 10.1186/1029-242X-2014-448. |




K and m, n | Methods | |
|||
Iter | fk (10) | CPU time (sec.) | |
||
K = 10 | Algorithm 1 | 3310 | 2.23e - 5 | 3.3438 | 0.0033 |
m = 210 | Lopez et al. [21] | 3491 | 3.23e - 5 | 3.8125 | 0.0045 |
n = 212 | Zhou and Wang [39] | 3991 | 2.79e - 4 | 4.7438 | 0.0012 |
K = 20 | Algorithm 1 | 7180 | 9.6601e - 13 | 5.08281 | 3.380e - 5 |
m = 210 | Lopez et al. [21] | 8901 | 7.6601e - 13 | 5.28281 | 3.271e - 5 |
n = 212 | Zhou and Wang [39] | 8180 | 8.4375e - 12 | 6.478 | 3.260e - 5 |
K = 50 | Algorithm 1 | 8780 | 5.7301e - 10 | 12.3312 | 4.276e - 4 |
m = 212 | Lopez et al. [21] | 9901 | 6.6601e - 9 | 11.2761 | 4.238e - 4 |
n = 213 | Zhou and Wang [39] | 9180 | 7.251e - 10 | 11.453 | 3.457e - 4 |
K and m, n | Methods | |
|||
Iter | fk (10) | CPU time (sec.) | |
||
K = 10 | Algorithm 1 | 3310 | 2.23e - 5 | 3.3438 | 0.0033 |
m = 210 | Lopez et al. [21] | 3491 | 3.23e - 5 | 3.8125 | 0.0045 |
n = 212 | Zhou and Wang [39] | 3991 | 2.79e - 4 | 4.7438 | 0.0012 |
K = 20 | Algorithm 1 | 7180 | 9.6601e - 13 | 5.08281 | 3.380e - 5 |
m = 210 | Lopez et al. [21] | 8901 | 7.6601e - 13 | 5.28281 | 3.271e - 5 |
n = 212 | Zhou and Wang [39] | 8180 | 8.4375e - 12 | 6.478 | 3.260e - 5 |
K = 50 | Algorithm 1 | 8780 | 5.7301e - 10 | 12.3312 | 4.276e - 4 |
m = 212 | Lopez et al. [21] | 9901 | 6.6601e - 9 | 11.2761 | 4.238e - 4 |
n = 213 | Zhou and Wang [39] | 9180 | 7.251e - 10 | 11.453 | 3.457e - 4 |
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