# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021160
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## An efficient iterative method for solving split variational inclusion problem with applications

 1 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand 2 Department of Mathematics, Faculty of Science, Usmanu Danfodiyo University, Sokoto 840244, Nigeria 3 Departments of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

* Corresponding author: poom.kum@kmutt.ac.th

Received  October 2020 Revised  May 2021 Early access September 2021

A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.

Citation: Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021160
##### 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 Analysis, 9 (2001), 3-11.  doi: 10.1023/A:1011253113155. [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons New York, 1984. [3] C. Baiocchi, Variational and quasivariational inequalities, Applications to Free-boundary Problems. [4] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7. [5] 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. [6] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-124.  doi: 10.1088/0266-5611/20/1/006. [7] C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal, 13 (2012), 759-775. [8] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine & Biology, 51 (2006), 2353.  doi: 10.1088/0031-9155/51/10/001. [9] 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. [10] 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. [11] T. Chamnarnpan, S. Phiangsungnoen and P. Kumam, A new hybrid extragradient algorithm for solving the equilibrium and variational inequality problems, Afrika Matematika, 26 (2015), 87-98.  doi: 10.1007/s13370-013-0187-x. [12] P. Cholamjiak, D. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Applicandae Mathematicae, 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7. [13] C.-S. Chuang, Hybrid inertial proximal algorithm for the split variational inclusion problem in hilbert spaces with applications, Optimization, 66 (2017), 777-792.  doi: 10.1080/02331934.2017.1306744. [14] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling & Simulation, 4 (2005), 1168-1200.  doi: 10.1137/050626090. [15] A. Gibali and D. V. Thong, Tseng type methods for solving inclusion problems and its applications, Calcolo, 55 (2018), Paper No. 49, 22 pp. doi: 10.1007/s10092-018-0292-1. [16] A. Gibali, D. V. Thong and N. T. Vinh, Three new iterative methods for solving inclusion problems and related problems, Computational and Applied Mathematics, 39 (2020), Paper No. 187, 23 pp. doi: 10.1007/s40314-020-01215-6. [17] A. Hanjing and S. Suantai, A fast image restoration algorithm based on a fixed point and optimization method, Mathematics, 8 (2020), 378.  doi: 10.3390/math8030378. [18] P. T. Harker, A variational inequality approach for the determination of oligopolistic market equilibrium, Mathematical Programming, 30 (1984), 105-111.  doi: 10.1007/BF02591802. [19] C. Jaiboon and P. Kumam, An extragradient approximation method for system of equilibrium problems and variational inequality problems, Thai Journal of Mathematics, 7 (2009), 77-104. [20] K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optimization Letters, 8 (2014), 1113-1124.  doi: 10.1007/s11590-013-0629-2. [21] E. N. Khobotov, Modification of the extra-gradient method for solving variational inequalities and certain optimization problems, USSR Computational Mathematics and Mathematical Physics, 27 (1987), 120-127.  doi: 10.1016/0041-5553(87)90058-9. [22] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, vol. 31, SIAM, 1980. [23] I. Konnov, Combined Relaxation Methods for Variational Inequalities, vol. 495, Springer Science & Business Media, 2001. doi: 10.1007/978-3-642-56886-2. [24] R. Kraikaew and S. Saejung, Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in hilbert spaces, Journal of Optimization Theory and Applications, 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2. [25] W. Kumam, H. Piri and P. Kumam, Solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings, Applied Mathematics and Computation, 248 (2014), 441-455.  doi: 10.1016/j.amc.2014.09.118. [26] P. Majee and C. Nahak, On inertial proximal algorithm for split variational inclusion problems, Optimization, 67 (2018), 1701-1716.  doi: 10.1080/02331934.2018.1486838. [27] Y. Malitsky, Golden ratio algorithms for variational inequalities, Mathematical Programming, 184 (2020), 383-410.  doi: 10.1007/s10107-019-01416-w. [28] P. Marcotte, Application of khobotov's algorithm to variational inequalities and network equilibrium problems, INFOR: Information Systems and Operational Research, 29 (1991), 258-270.  doi: 10.1080/03155986.1991.11732174. [29] G. Marino and H.-K. Xu, Convergence of generalized proximal point algorithms, Commun. Pure Appl. Anal, 3 (2004), 791-808.  doi: 10.3934/cpaa.2004.3.791. [30] A. Moudafi, Split monotone variational inclusions, Journal of Optimization Theory and Applications, 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6. [31] M. A. Noor, Some developments in general variational inequalities, Applied Mathematics and Computation, 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7. [32] E. U. Ofoedu, Strong convergence theorem for uniformly l-lipschitzian asymptotically pseudocontractive mapping in real banach space, Journal of Mathematical Analysis and Applications, 321 (2006), 722-728.  doi: 10.1016/j.jmaa.2005.08.076. [33] W. Takahashi, Nonlinear functional analysis-fixed point theory and its applications, 2000. [34] D. V. Thong, V. T. Dung and Y. J. Cho, A new strong convergence for solving split variational inclusion problems, Numer Algorithms, 86 (2021), 565-591.  doi: 10.1007/s11075-020-00901-0. [35] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861. [36] L. Yang and F. H. Zhao, General split variational inclusion problem in hilbert spaces, Abstract and Applied Analysis, vol. 2014, Art. ID 816035, 8 pp. doi: 10.1155/2014/816035.

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 Analysis, 9 (2001), 3-11.  doi: 10.1023/A:1011253113155. [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons New York, 1984. [3] C. Baiocchi, Variational and quasivariational inequalities, Applications to Free-boundary Problems. [4] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7. [5] 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. [6] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-124.  doi: 10.1088/0266-5611/20/1/006. [7] C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal, 13 (2012), 759-775. [8] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine & Biology, 51 (2006), 2353.  doi: 10.1088/0031-9155/51/10/001. [9] 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. [10] 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. [11] T. Chamnarnpan, S. Phiangsungnoen and P. Kumam, A new hybrid extragradient algorithm for solving the equilibrium and variational inequality problems, Afrika Matematika, 26 (2015), 87-98.  doi: 10.1007/s13370-013-0187-x. [12] P. Cholamjiak, D. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Applicandae Mathematicae, 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7. [13] C.-S. Chuang, Hybrid inertial proximal algorithm for the split variational inclusion problem in hilbert spaces with applications, Optimization, 66 (2017), 777-792.  doi: 10.1080/02331934.2017.1306744. [14] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling & Simulation, 4 (2005), 1168-1200.  doi: 10.1137/050626090. [15] A. Gibali and D. V. Thong, Tseng type methods for solving inclusion problems and its applications, Calcolo, 55 (2018), Paper No. 49, 22 pp. doi: 10.1007/s10092-018-0292-1. [16] A. Gibali, D. V. Thong and N. T. Vinh, Three new iterative methods for solving inclusion problems and related problems, Computational and Applied Mathematics, 39 (2020), Paper No. 187, 23 pp. doi: 10.1007/s40314-020-01215-6. [17] A. Hanjing and S. Suantai, A fast image restoration algorithm based on a fixed point and optimization method, Mathematics, 8 (2020), 378.  doi: 10.3390/math8030378. [18] P. T. Harker, A variational inequality approach for the determination of oligopolistic market equilibrium, Mathematical Programming, 30 (1984), 105-111.  doi: 10.1007/BF02591802. [19] C. Jaiboon and P. Kumam, An extragradient approximation method for system of equilibrium problems and variational inequality problems, Thai Journal of Mathematics, 7 (2009), 77-104. [20] K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optimization Letters, 8 (2014), 1113-1124.  doi: 10.1007/s11590-013-0629-2. [21] E. N. Khobotov, Modification of the extra-gradient method for solving variational inequalities and certain optimization problems, USSR Computational Mathematics and Mathematical Physics, 27 (1987), 120-127.  doi: 10.1016/0041-5553(87)90058-9. [22] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, vol. 31, SIAM, 1980. [23] I. Konnov, Combined Relaxation Methods for Variational Inequalities, vol. 495, Springer Science & Business Media, 2001. doi: 10.1007/978-3-642-56886-2. [24] R. Kraikaew and S. Saejung, Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in hilbert spaces, Journal of Optimization Theory and Applications, 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2. [25] W. Kumam, H. Piri and P. Kumam, Solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings, Applied Mathematics and Computation, 248 (2014), 441-455.  doi: 10.1016/j.amc.2014.09.118. [26] P. Majee and C. Nahak, On inertial proximal algorithm for split variational inclusion problems, Optimization, 67 (2018), 1701-1716.  doi: 10.1080/02331934.2018.1486838. [27] Y. Malitsky, Golden ratio algorithms for variational inequalities, Mathematical Programming, 184 (2020), 383-410.  doi: 10.1007/s10107-019-01416-w. [28] P. Marcotte, Application of khobotov's algorithm to variational inequalities and network equilibrium problems, INFOR: Information Systems and Operational Research, 29 (1991), 258-270.  doi: 10.1080/03155986.1991.11732174. [29] G. Marino and H.-K. Xu, Convergence of generalized proximal point algorithms, Commun. Pure Appl. Anal, 3 (2004), 791-808.  doi: 10.3934/cpaa.2004.3.791. [30] A. Moudafi, Split monotone variational inclusions, Journal of Optimization Theory and Applications, 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6. [31] M. A. Noor, Some developments in general variational inequalities, Applied Mathematics and Computation, 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7. [32] E. U. Ofoedu, Strong convergence theorem for uniformly l-lipschitzian asymptotically pseudocontractive mapping in real banach space, Journal of Mathematical Analysis and Applications, 321 (2006), 722-728.  doi: 10.1016/j.jmaa.2005.08.076. [33] W. Takahashi, Nonlinear functional analysis-fixed point theory and its applications, 2000. [34] D. V. Thong, V. T. Dung and Y. J. Cho, A new strong convergence for solving split variational inclusion problems, Numer Algorithms, 86 (2021), 565-591.  doi: 10.1007/s11075-020-00901-0. [35] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861. [36] L. Yang and F. H. Zhao, General split variational inclusion problem in hilbert spaces, Abstract and Applied Analysis, vol. 2014, Art. ID 816035, 8 pp. doi: 10.1155/2014/816035.
Comparative results for random instances.
Results of the compared algorithms with different cases of initial points.
Original test images of Monarch, Flowers and Colorchecker.
Original cropped test images of Monarch, Flowers and Colorchecker.
Degraded and restored Monarch images by the compared algorithms.
Degraded and restored cropped Monarch images by the compared algorithms.
Degraded and restored Flowers images by the compared algorithms.
Degraded and restored cropped Flowers images by the compared algorithms.
Degraded and restored Colorchecker images by the compared algorithms.
Degraded and restored cropped Colorchecker images by the compared algorithms.
The PNSR and SSIM values of the compared algorithms
 Scheme 51 Algorithm 4.4 Images SNR SSIM SNR SSIM Monarch 43.3255 0.9684 39.5788 0.9624 Flowers 40.6001 0.9116 36.7745 0.8660 Colorchecker 41.7454 0.8996 39.0134 0.9061
 Scheme 51 Algorithm 4.4 Images SNR SSIM SNR SSIM Monarch 43.3255 0.9684 39.5788 0.9624 Flowers 40.6001 0.9116 36.7745 0.8660 Colorchecker 41.7454 0.8996 39.0134 0.9061
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