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April  2019, 15(2): 963-984. doi: 10.3934/jimo.2018080

A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications

Aviv Gibali 1,2,, , Dang Thi Mai 3, and  Nguyen The Vinh 3,

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

* Corresponding author: avivg@braude.ac.il

Received  November 2017 Revised  January 2018 Published  April 2019 Early access  June 2018

Fund Project: The first author work is supported by the EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669. The research of the third author was partially supported by University of Transport and Communications (UTC) [grant number T2018-CB-002] and Vietnam Institute for Advanced Study in Mathematics (VIASM).

Inspired by the works of López et al. [21] and the recent paper of Dang et al. [15], we devise a new inertial relaxation of the CQ algorithm for solving Split Feasibility Problems (SFP) in real Hilbert spaces. Under mild and standard conditions we establish weak convergence of the proposed algorithm. We also propose a Mann-type variant which converges strongly. The performances and comparisons with some existing methods are presented through numerical examples in Compressed Sensing and Sparse Binary Tomography by solving the LASSO problem.

Keywords: Split feasibility problem, CQ algorithm, inertial technique, self-adaptive algorithm, weak convergence, strong convergence, LASSO problem.
Mathematics Subject Classification: Primary: 65K05, 65K10; Secondary: 49J52.
Citation: 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 & Management Optimization, 2019, 15 (2) : 963-984. doi: 10.3934/jimo.2018080
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.  Google Scholar

[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.  Google Scholar

[3]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Chen, D. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Comput., 20 (1998), 33-61.  doi: 10.1137/S1064827596304010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

Y. Lou and M. Yan, Fast l1 - l2 Minimization via a proximal operator, arXiv: 1609.09530. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Tibshirani, Regression shrinkage and selection Via the lasso, J. Royal Stat. Soc., 58 (1996), 267-288.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

H.-K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Chen, D. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Comput., 20 (1998), 33-61.  doi: 10.1137/S1064827596304010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

Y. Lou and M. Yan, Fast l1 - l2 Minimization via a proximal operator, arXiv: 1609.09530. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Tibshirani, Regression shrinkage and selection Via the lasso, J. Royal Stat. Soc., 58 (1996), 267-288.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

H.-K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Numerical results for m = 210; n = 212; k = 20
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Figure 2.  Numerical results for m = 210; n = 212; k = 40
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Figure 3.  Numerical results for m = 212; n = 213; k = 50
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Figure 4.  Parallelbeam geometry set-up: a set of parallel rays is shot through the object from different directions. These are typically coined as one projection. Two projections are illustrated above. (Left) Illustration of a single projection corresponding to a measurement along one ray. The image domain $\Omega$ is tiled into pixels or mathematically Haar-basis functions. Hence, a single projection corresponds to the line integral over a piecewise constant function
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4] (left). We illustrate how such a $32\times 32$ image is sampled (right) along $45$ parallel and equidistant lines that are all perpendicular to $\theta: = (\cos(30^\circ),\sin(30^\circ))^T$">Figure 5.  Vessel test image from [4] (left). We illustrate how such a $32\times 32$ image is sampled (right) along $45$ parallel and equidistant lines that are all perpendicular to $\theta: = (\cos(30^\circ),\sin(30^\circ))^T$
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4] representing a vascular system containing larger and smaller vessels. The results are for the recover $u$ from a $15$ (limited number of) tomographic projections">Figure 6.  Reconstructing a binary test image $u\in \mathbb{R}^{64\times 64}$ from [4] representing a vascular system containing larger and smaller vessels. The results are for the recover $u$ from a $15$ (limited number of) tomographic projections
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Table 1.  Numerical results obtained by Algorithm 1 compared with Lopez et al. algorithm [21] ((8)-(10)) and Zhou and Wang [39,Algorithm 3.1]
K and m, n Methods $\epsilon $=10-6
Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$
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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K and m, n Methods $\epsilon $=10-6
Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$
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
Table 2.  Numerical results obtained by Algorithm 1 compared with Lopez et al. algorithm [21] ((8)-(9)) and Zhou and Wang [39,Algorithm 3.1]
K and m, n Methods $ \epsilon$ = 10-6
Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$
K = 916 Algorithm 1 87 37.6590 0.6406 65.7008
m = 1365 Lopez et al. [21] 100 49.3198 0.3541 38.2665
n = 4096 Zhou and Wang [39] 100 48.5597 0.6565 68.3321
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K and m, n Methods $ \epsilon$ = 10-6
Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$
K = 916 Algorithm 1 87 37.6590 0.6406 65.7008
m = 1365 Lopez et al. [21] 100 49.3198 0.3541 38.2665
n = 4096 Zhou and Wang [39] 100 48.5597 0.6565 68.3321
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