doi: 10.3934/jimo.2021185
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New iterative regularization methods for solving split variational inclusion problems

1. 

TIMAS - Thang Long University, Ha Noi, Vietnam

2. 

Institute of Mathematics, VAST, Hanoi, 18 Hoang Quoc Viet, Hanoi, Vietnam

3. 

Department of Basic Sciences, College of Air Force, Nha Trang City, Vietnam

* Corresponding author: Dang Van Hieu (dangvanhieu@tdtu.edu.vn)

Received  September 2020 Revised  March 2021 Early access November 2021

The paper proposes some new iterative algorithms for solving a split variational inclusion problem involving maximally monotone multi-valued operators in a Hilbert space. The algorithms are constructed around the resolvent of operator and the regularization technique to get the strong convergence. Some stepsize rules are incorporated to allow the algorithms to work easily. An application of the proposed algorithms to split feasibility problems is also studied. The computational performance of the new algorithms in comparison with others is shown by some numerical experiments.

Citation: Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021185
References:
[1]

P. K. Anh and D. V. Hieu, Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems, Vietnam J. Math., 44 (2016), 351-374.  doi: 10.1007/s10013-015-0129-z.  Google Scholar

[2]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problems, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[3]

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

[4]

C. ByrneY. Censor and A. Gibali, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[5]

H. Brézis and I. I. Chapitre, Opérateurs maximaux monotones, North-Holland Math. Stud., 5 (1973), 19-51.   Google Scholar

[6]

C-S. Chuang, Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., 2013 (2013), 20pp. doi: 10.1186/1687-1812-2013-350.  Google Scholar

[7]

C-S. Chuang, Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem, Optimization, 65 (2016), 859-876.  doi: 10.1080/02331934.2015.1072715.  Google Scholar

[8]

P. Cholamjiak, D. V. Hieu and Y. J. Cho, Relaxed forward-backward splitting methods for solving variational inclusions and applications, J. Sci. Comput., 88 (2021), 23pp. doi: 10.1007/s10915-021-01608-7.  Google Scholar

[9]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.  Google Scholar

[10]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.   Google Scholar

[11]

Y. Censor and T. Elfving, A multiprojections algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[12]

Y. CensorT. ElfvingN. 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]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[14]

Y. CensorA. Gibali and S. Reich, A von Neumann alternating method for finding common solutions to variational inequalities, Nonlinear Anal., 75 (2012), 4596-4603.  doi: 10.1016/j.na.2012.01.021.  Google Scholar

[15]

Y. CensorA. GibaliS. Reich and S. Sabach, Common solutions to variational inequalities, Set. Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x.  Google Scholar

[16]

Y. Censor and A. Segal, Iterative projection methods in biomedical inverse problems. In: Censor Y, Jiang M, Louis AK (eds) Mathematical methods in biomedical imaging and intensity-modulated therapy, IMRT, CRM Series, Ed. Norm., Pisa, 7 (2008), 65-96.   Google Scholar

[17]

B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert spaces, Numer. Funct. Anal. Optim., 13 (1992), 413-429.  doi: 10.1080/01630569208816489.  Google Scholar

[18]

D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478-501.  doi: 10.3846/13926292.2016.1183527.  Google Scholar

[19]

D. V. Hieu, Two hybrid algorithms for solving split equilibrium problems, Int. J. Comput. Math., 95 (2018), 561-583.  doi: 10.1080/00207160.2017.1291934.  Google Scholar

[20]

D. V. Hieu, Projection methods for solving split equilibrium problems, J. Ind. Manag. Optim., 16 (2020), 2331-2349.  doi: 10.3934/jimo.2019056.  Google Scholar

[21]

D. V. Hieu, P. K. Anh and N. H. Ha, Regularization proximal method for monotone variational inclusions, Netw. Spat. Econ., 2021. doi: 10.1007/s11067-021-09552-7.  Google Scholar

[22]

D. V. HieuP. K. Anh and L. D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., 73 (2019), 913-932.  doi: 10.1007/s10589-019-00093-x.  Google Scholar

[23]

D. V. Hieu, P. K. Anh, L. D. Muu and J. J. Strodiot, Iterative regularization methods with new stepsize rules for solving variational inclusions, J. Appl. Math. Comput., 2021. doi: 10.1007/s12190-021-01534-9.  Google Scholar

[24]

D. V. HieuY. J. ChoY-B. Xiao and P. Kumam, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., 49 (2021), 1165-1183.  doi: 10.1007/s10013-020-00447-7.  Google Scholar

[25]

D. V. Hieu, S. Reich, P. K. Anh and N. H. Ha, A new proximal-like algorithm for solving split variational inclusion problems, Numer. Algor., 2021. doi: 10.1007/s11075-021-01135-4.  Google Scholar

[26]

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction, Mathematics and its Applications, 52. Kluwer Academic Publishers Group, Dordrecht, 1989.  Google Scholar

[27]

L. V. LongD. V. Thong and V. T. Dung, New algorithms for the split variational inclusion problems and application to split feasibility problems, Optimization, 68 (2019), 2335-2363.  doi: 10.1080/02331934.2019.1631821.  Google Scholar

[28]

A. Moudafi, Split monotone variational inclusions, J Optim Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.  Google Scholar

[29]

A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615.  Google Scholar

[30]

A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett., 8 (2014), 2099-2110.  doi: 10.1007/s11590-013-0708-4.  Google Scholar

[31] H. Stark, Image Recovery: Theory and Applications, Academic Press, Orlando, FL, 1987.   Google Scholar
[32]

J. J. StrodiotD. M. Giang and V. H. Nguyen, Strong convergence of an iterative method for solving the multiple-set split equality fixed point problem in a real Hilbert space, Rev. R. Acad. Cienc. Exactas Fas. Nat. Ser. A Mat. RACSAM, 111 (2017), 983-998.  doi: 10.1007/s13398-016-0338-7.  Google Scholar

[33]

J. J. StrodiotP. T. Vuong and V. H. Nguyen, A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces, Optimization, 64 (2015), 2321-2341.  doi: 10.1080/02331934.2014.967237.  Google Scholar

[34]

K. SitthithakerngkietJ. Deepho and P. Kumam, A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems, Appl. Math. Comput., 250 (2015), 986-1001.  doi: 10.1016/j.amc.2014.10.130.  Google Scholar

[35]

S. TakahashiW. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27-41.  doi: 10.1007/s10957-010-9713-2.  Google Scholar

[36]

H. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.  doi: 10.1017/S0004972700020116.  Google Scholar

show all references

References:
[1]

P. K. Anh and D. V. Hieu, Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems, Vietnam J. Math., 44 (2016), 351-374.  doi: 10.1007/s10013-015-0129-z.  Google Scholar

[2]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problems, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[3]

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

[4]

C. ByrneY. Censor and A. Gibali, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[5]

H. Brézis and I. I. Chapitre, Opérateurs maximaux monotones, North-Holland Math. Stud., 5 (1973), 19-51.   Google Scholar

[6]

C-S. Chuang, Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., 2013 (2013), 20pp. doi: 10.1186/1687-1812-2013-350.  Google Scholar

[7]

C-S. Chuang, Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem, Optimization, 65 (2016), 859-876.  doi: 10.1080/02331934.2015.1072715.  Google Scholar

[8]

P. Cholamjiak, D. V. Hieu and Y. J. Cho, Relaxed forward-backward splitting methods for solving variational inclusions and applications, J. Sci. Comput., 88 (2021), 23pp. doi: 10.1007/s10915-021-01608-7.  Google Scholar

[9]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.  Google Scholar

[10]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.   Google Scholar

[11]

Y. Censor and T. Elfving, A multiprojections algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[12]

Y. CensorT. ElfvingN. 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]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[14]

Y. CensorA. Gibali and S. Reich, A von Neumann alternating method for finding common solutions to variational inequalities, Nonlinear Anal., 75 (2012), 4596-4603.  doi: 10.1016/j.na.2012.01.021.  Google Scholar

[15]

Y. CensorA. GibaliS. Reich and S. Sabach, Common solutions to variational inequalities, Set. Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x.  Google Scholar

[16]

Y. Censor and A. Segal, Iterative projection methods in biomedical inverse problems. In: Censor Y, Jiang M, Louis AK (eds) Mathematical methods in biomedical imaging and intensity-modulated therapy, IMRT, CRM Series, Ed. Norm., Pisa, 7 (2008), 65-96.   Google Scholar

[17]

B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert spaces, Numer. Funct. Anal. Optim., 13 (1992), 413-429.  doi: 10.1080/01630569208816489.  Google Scholar

[18]

D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478-501.  doi: 10.3846/13926292.2016.1183527.  Google Scholar

[19]

D. V. Hieu, Two hybrid algorithms for solving split equilibrium problems, Int. J. Comput. Math., 95 (2018), 561-583.  doi: 10.1080/00207160.2017.1291934.  Google Scholar

[20]

D. V. Hieu, Projection methods for solving split equilibrium problems, J. Ind. Manag. Optim., 16 (2020), 2331-2349.  doi: 10.3934/jimo.2019056.  Google Scholar

[21]

D. V. Hieu, P. K. Anh and N. H. Ha, Regularization proximal method for monotone variational inclusions, Netw. Spat. Econ., 2021. doi: 10.1007/s11067-021-09552-7.  Google Scholar

[22]

D. V. HieuP. K. Anh and L. D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., 73 (2019), 913-932.  doi: 10.1007/s10589-019-00093-x.  Google Scholar

[23]

D. V. Hieu, P. K. Anh, L. D. Muu and J. J. Strodiot, Iterative regularization methods with new stepsize rules for solving variational inclusions, J. Appl. Math. Comput., 2021. doi: 10.1007/s12190-021-01534-9.  Google Scholar

[24]

D. V. HieuY. J. ChoY-B. Xiao and P. Kumam, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., 49 (2021), 1165-1183.  doi: 10.1007/s10013-020-00447-7.  Google Scholar

[25]

D. V. Hieu, S. Reich, P. K. Anh and N. H. Ha, A new proximal-like algorithm for solving split variational inclusion problems, Numer. Algor., 2021. doi: 10.1007/s11075-021-01135-4.  Google Scholar

[26]

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction, Mathematics and its Applications, 52. Kluwer Academic Publishers Group, Dordrecht, 1989.  Google Scholar

[27]

L. V. LongD. V. Thong and V. T. Dung, New algorithms for the split variational inclusion problems and application to split feasibility problems, Optimization, 68 (2019), 2335-2363.  doi: 10.1080/02331934.2019.1631821.  Google Scholar

[28]

A. Moudafi, Split monotone variational inclusions, J Optim Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.  Google Scholar

[29]

A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615.  Google Scholar

[30]

A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett., 8 (2014), 2099-2110.  doi: 10.1007/s11590-013-0708-4.  Google Scholar

[31] H. Stark, Image Recovery: Theory and Applications, Academic Press, Orlando, FL, 1987.   Google Scholar
[32]

J. J. StrodiotD. M. Giang and V. H. Nguyen, Strong convergence of an iterative method for solving the multiple-set split equality fixed point problem in a real Hilbert space, Rev. R. Acad. Cienc. Exactas Fas. Nat. Ser. A Mat. RACSAM, 111 (2017), 983-998.  doi: 10.1007/s13398-016-0338-7.  Google Scholar

[33]

J. J. StrodiotP. T. Vuong and V. H. Nguyen, A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces, Optimization, 64 (2015), 2321-2341.  doi: 10.1080/02331934.2014.967237.  Google Scholar

[34]

K. SitthithakerngkietJ. Deepho and P. Kumam, A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems, Appl. Math. Comput., 250 (2015), 986-1001.  doi: 10.1016/j.amc.2014.10.130.  Google Scholar

[35]

S. TakahashiW. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27-41.  doi: 10.1007/s10957-010-9713-2.  Google Scholar

[36]

H. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.  doi: 10.1017/S0004972700020116.  Google Scholar

Figure 1.  Example 1 for $ M = 100, N = 200 $. Number of iteration is 1393, 1395, 1365, 1414, 1402, respectively
Figure 2.  Example 1 for M = 200, N = 500. Number of iteration is 202, 204, 198, 209, 210, respectively
Figure 3.  Example 2 for M = 1024, N = 256. Number of iteration is 763, 759, 228, 748, 756, respectively
Figure 4.  Example 2 for M = 2048, N = 512. Number of iteration is 357,358,171,359,353, respectively
Figure 5.  Example 3 for x0(t) = 1. Number of iteration is 350,361,189,414,340, respectively
Figure 6.  Example 3 for x0(t) = exp(−t). Number of iteration is 238,217,119,335,283, respectively
Table1 
Algorithm 1:
Initialization: Take $ x_0 \in \mathcal{H}_1 $ and two sequences $ \left\{\lambda_n\right\},\,\left\{\alpha_n\right\} \subset \left(0,+\infty\right) $.
Iterative Steps:
    Step 1. Compute $ x_{n+1}=J_{\lambda_n\mathcal{B}_1}\left(x_n-\lambda_n\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\lambda_n\alpha_n\mathcal{F}x_n\right). $
    Step 2. Set $ n:=n+1 $ and return to Step 1.
Algorithm 1:
Initialization: Take $ x_0 \in \mathcal{H}_1 $ and two sequences $ \left\{\lambda_n\right\},\,\left\{\alpha_n\right\} \subset \left(0,+\infty\right) $.
Iterative Steps:
    Step 1. Compute $ x_{n+1}=J_{\lambda_n\mathcal{B}_1}\left(x_n-\lambda_n\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\lambda_n\alpha_n\mathcal{F}x_n\right). $
    Step 2. Set $ n:=n+1 $ and return to Step 1.
Table2 
Algorithm 2:
Initialization: Take $ x_0 \in \mathcal{H}_1 $ and $ \lambda_0>0 $, $ \mu \in (0,1) $. Choose a sequence $ \left\{\alpha_n\right\} \subset \left(0,+\infty\right) $ such that conditions $ \rm (C2) - (C4) $ hold.
Iterative Steps:
    Step 1. Compute
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\{ \begin{array}{ll} y_n = J_{\lambda_n\mathcal{B}_1}\left(x_n-\lambda_n\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\lambda_n\alpha_n\mathcal{F}x_n\right).\\ x_{n+1} = y_n+\lambda_n\left(\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n\right). \end{array} \right. $
    Step 2. Update $ \lambda_n $:
$ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \lambda_{n+1} = \min \left\{\lambda_n,\frac{\mu ||x_n-y_n||}{||\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n||}\right\}.$
Algorithm 2:
Initialization: Take $ x_0 \in \mathcal{H}_1 $ and $ \lambda_0>0 $, $ \mu \in (0,1) $. Choose a sequence $ \left\{\alpha_n\right\} \subset \left(0,+\infty\right) $ such that conditions $ \rm (C2) - (C4) $ hold.
Iterative Steps:
    Step 1. Compute
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\{ \begin{array}{ll} y_n = J_{\lambda_n\mathcal{B}_1}\left(x_n-\lambda_n\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\lambda_n\alpha_n\mathcal{F}x_n\right).\\ x_{n+1} = y_n+\lambda_n\left(\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n\right). \end{array} \right. $
    Step 2. Update $ \lambda_n $:
$ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \lambda_{n+1} = \min \left\{\lambda_n,\frac{\mu ||x_n-y_n||}{||\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}x_n-\mathcal{A}^*(I-J_{\mathcal{B}_2})\mathcal{A}y_n||}\right\}.$
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