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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 and 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. [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. [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. [4] C. Byrne, Y. 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. [5] H. Brézis and I. I. Chapitre, Opérateurs maximaux monotones, North-Holland Math. Stud., 5 (1973), 19-51. [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. [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. [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. [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. [10] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. [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. [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] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5. [14] Y. Censor, A. 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. [15] Y. Censor, A. Gibali, S. Reich and S. Sabach, Common solutions to variational inequalities, Set. Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x. [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. [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. [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. [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. [20] D. V. Hieu, Projection methods for solving split equilibrium problems, J. Ind. Manag. Optim., 16 (2020), 2331-2349.  doi: 10.3934/jimo.2019056. [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. [22] D. V. Hieu, P. 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. [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. [24] D. V. Hieu, Y. J. Cho, Y-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. [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. [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. [27] L. V. Long, D. 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. [28] A. Moudafi, Split monotone variational inclusions, J Optim Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6. [29] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615. [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. [31] H. Stark, Image Recovery: Theory and Applications, Academic Press, Orlando, FL, 1987. [32] J. J. Strodiot, D. 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. [33] J. J. Strodiot, P. 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. [34] K. Sitthithakerngkiet, J. 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. [35] S. Takahashi, W. 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. [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.

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. [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. [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. [4] C. Byrne, Y. 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. [5] H. Brézis and I. I. Chapitre, Opérateurs maximaux monotones, North-Holland Math. Stud., 5 (1973), 19-51. [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. [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. [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. [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. [10] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. [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. [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] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5. [14] Y. Censor, A. 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. [15] Y. Censor, A. Gibali, S. Reich and S. Sabach, Common solutions to variational inequalities, Set. Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x. [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. [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. [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. [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. [20] D. V. Hieu, Projection methods for solving split equilibrium problems, J. Ind. Manag. Optim., 16 (2020), 2331-2349.  doi: 10.3934/jimo.2019056. [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. [22] D. V. Hieu, P. 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. [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. [24] D. V. Hieu, Y. J. Cho, Y-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. [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. [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. [27] L. V. Long, D. 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. [28] A. Moudafi, Split monotone variational inclusions, J Optim Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6. [29] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615. [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. [31] H. Stark, Image Recovery: Theory and Applications, Academic Press, Orlando, FL, 1987. [32] J. J. Strodiot, D. 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. [33] J. J. Strodiot, P. 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. [34] K. Sitthithakerngkiet, J. 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. [35] S. Takahashi, W. 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. [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.
Example 1 for $M = 100, N = 200$. Number of iteration is 1393, 1395, 1365, 1414, 1402, respectively
Example 1 for M = 200, N = 500. Number of iteration is 202, 204, 198, 209, 210, respectively
Example 2 for M = 1024, N = 256. Number of iteration is 763, 759, 228, 748, 756, respectively
Example 2 for M = 2048, N = 512. Number of iteration is 357,358,171,359,353, respectively
Example 3 for x0(t) = 1. Number of iteration is 350,361,189,414,340, respectively
Example 3 for x0(t) = exp(−t). Number of iteration is 238,217,119,335,283, respectively
 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.
 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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