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 & Management Optimization, doi: 10.3934/jimo.2021160
References:
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T. ChamnarnpanS. 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.  Google Scholar

[12]

P. CholamjiakD. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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I. Konnov, Combined Relaxation Methods for Variational Inequalities, vol. 495, Springer Science & Business Media, 2001. doi: 10.1007/978-3-642-56886-2.  Google Scholar

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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.  Google Scholar

[25]

W. KumamH. 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.  Google Scholar

[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.  Google Scholar

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Y. Malitsky, Golden ratio algorithms for variational inequalities, Mathematical Programming, 184 (2020), 383-410.  doi: 10.1007/s10107-019-01416-w.  Google Scholar

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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.  Google Scholar

[29]

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[30]

A. Moudafi, Split monotone variational inclusions, Journal of Optimization Theory and Applications, 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.  Google Scholar

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[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.  Google Scholar

[33]

W. Takahashi, Nonlinear functional analysis-fixed point theory and its applications, 2000.  Google Scholar

[34]

D. V. ThongV. 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.  Google Scholar

[35]

Z. WangA. C. BovikH. 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.  Google Scholar

[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.  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 Analysis, 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar

[2]

J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons New York, 1984.  Google Scholar

[3]

C. Baiocchi, Variational and quasivariational inequalities, Applications to Free-boundary Problems. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. ByrneY. CensorA. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal, 13 (2012), 759-775.   Google Scholar

[8]

Y. CensorT. BortfeldB. 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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

T. ChamnarnpanS. 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.  Google Scholar

[12]

P. CholamjiakD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, vol. 31, SIAM, 1980. Google Scholar

[23]

I. Konnov, Combined Relaxation Methods for Variational Inequalities, vol. 495, Springer Science & Business Media, 2001. doi: 10.1007/978-3-642-56886-2.  Google Scholar

[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.  Google Scholar

[25]

W. KumamH. 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.  Google Scholar

[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.  Google Scholar

[27]

Y. Malitsky, Golden ratio algorithms for variational inequalities, Mathematical Programming, 184 (2020), 383-410.  doi: 10.1007/s10107-019-01416-w.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

A. Moudafi, Split monotone variational inclusions, Journal of Optimization Theory and Applications, 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

W. Takahashi, Nonlinear functional analysis-fixed point theory and its applications, 2000.  Google Scholar

[34]

D. V. ThongV. 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.  Google Scholar

[35]

Z. WangA. C. BovikH. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Comparative results for random instances.
Figure 2.  Results of the compared algorithms with different cases of initial points.
Figure 3.  Original test images of Monarch, Flowers and Colorchecker.
Figure 4.  Original cropped test images of Monarch, Flowers and Colorchecker.
Figure 5.  Degraded and restored Monarch images by the compared algorithms.
Figure 6.  Degraded and restored cropped Monarch images by the compared algorithms.
Figure 7.  Degraded and restored Flowers images by the compared algorithms.
Figure 8.  Degraded and restored cropped Flowers images by the compared algorithms.
Figure 9.  Degraded and restored Colorchecker images by the compared algorithms.
Figure 10.  Degraded and restored cropped Colorchecker images by the compared algorithms.
Table 1.  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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