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A strongly convergent algorithm for solving common variational inclusion with application to image recovery problems. (English) Zbl 1484.65113

Summary: In this paper, we present an accelerated approach for addressing the common variational inclusion problem in real Hilbert spaces that incorporates several techniques. Under conventional and appropriate assumptions, the algorithm’s strong convergence theorem is established, and the applicability and benefits of the innovative approach for image recovery problems are demonstrated.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: DOI

References:

[1] Anh, P. K.; Hieu, D. V., Parallel and sequential hybrid methods for a finite family of asymptotically quasi ϕ-nonexpansive mappings, J. Appl. Math. Comput., 48, 1, 241-263 (2015) · Zbl 1325.47128
[2] Anh, P. K.; Hieu, D. V., Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems, Vietnam J. Math., 44, 2, 351-374 (2016) · Zbl 1347.47038
[3] Abubakar, J.; Kumam, P.; Garba, A. I.; Abdullahi, M. S.; Ibrahim, A. H.; Sitthithakerngkiet, K., An inertial iterative scheme for solving variational inclusion with application to Nash-Cournot equilibrium and image restoration problems, Carpath. J. Math., 37, 3, 361-380 (2021) · Zbl 1485.47087
[4] Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 1, 183-202 (2009) · Zbl 1175.94009
[5] Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C., A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21, 93-108 (2020) · Zbl 1477.47060
[6] Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C., Pseudomonotone variational inequalities and fixed points, Fixed Point Theory, 22, 543-558 (2021) · Zbl 1489.47088
[7] Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C., Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70, 1337-1358 (2021) · Zbl 1486.47105
[8] Ceng, L. C.; Shang, M., Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70, 715-740 (2021) · Zbl 07339862
[9] Ceng, L. C.; Yuan, Q., Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems, J. Inequal. Appl., Article 274 pp. (2019) · Zbl 1499.47024
[10] Cholamjiak, W.; Cholamjiak, P.; Suantai, S., An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces, J. Fixed Point Theory Appl., 20, 1, 1-17 (2018) · Zbl 1491.47059
[11] Cholamjiak, W.; Khan, S. A.; Yambangwai, D.; Kazmi, K. R., Strong convergence analysis of common variational inclusion problems involving an inertial parallel monotone hybrid method for a novel application to image restoration, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 114, 2 (2020) · Zbl 1443.47055
[12] Cholamjiak, P.; Hieu, D. V.; Cho, Y. J., Relaxed forward-backward splitting methods for solving variational inclusions and applications, J. Sci. Comput., 88, 3 (2021) · Zbl 1490.65100
[13] Combettes, P. L.; Wajs, V. R., Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4, 4, 1168-1200 (2005) · Zbl 1179.94031
[14] Dixit, A.; Sahu, D. R.; Singh, A. K.; Som, T., Application of a new accelerated algorithm to regression problems, Soft Comput., 24, 2, 1539-1552 (2020) · Zbl 1436.65069
[15] Gibali, A.; Thong, D. V., Tseng type methods for solving inclusion problems and its applications, Calcolo, 55, 4, Article 49 pp. (2018) · Zbl 1482.65096
[16] Goeleven, D., Complementarity and Variational Inequalities in Electronics (2017), Academic Press, Elsevier · Zbl 1377.94001
[17] He, L.; Cui, Y. L.; Ceng, L. C.; Zhao, T. Y.; Wang, D. Q.; Hu, H. Y., Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule, J. Inequal. Appl., Article 146 pp. (2021) · Zbl 1504.49016
[18] Hieu, D. V.; Muu, L. D.; Anh, P. K., Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms, 73, 1, 197-217 (2016) · Zbl 1367.65089
[19] Hieu, D. V., Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings, J. Appl. Math. Comput., 53, 1-2, 531-554 (2017) · Zbl 1364.65117
[20] Nash, John F.; Rassias, M. Th., Open Problems in Mathematics (2016), Springer · Zbl 1351.00027
[21] Lions, P.-L.; Mercier, B., Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16, 6, 964-979 (1979) · Zbl 0426.65050
[22] Liu, H.; Yang, J., Weak convergence of iterative methods for solving quasimonotone variational inequalities, Comput. Optim. Appl., 77, 491-508 (2020) · Zbl 07342388
[23] Lorenz, D.; Pock, T., An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51, 311-325 (2015) · Zbl 1327.47063
[24] Martinez-Yanes, C.; Xu, H. K., Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal., 64, 11, 2400-2411 (2006) · Zbl 1105.47060
[25] Motreanu, D., Nonlinear Differential Problems with Smooth and Nonsmooth Constraints (2018), Academic Press, Elsevier · Zbl 1403.35006
[26] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 2, 372-379 (2003) · Zbl 1035.47048
[27] Noor, M. A.; Noor, K. I.; Rassias, M. Th., New trends in general variational inequalities, Acta Appl. Math., 170, 1, 981-1064 (2020) · Zbl 1460.49007
[28] Nesterov, Y., A method for solving the convex programming problem with convergence rate \(O(1 / k^2)\), Dokl. Akad. Nauk SSSR, 269, 543-547 (1983) · Zbl 0535.90071
[29] O’Donoghue, B.; Candès, E. J., Adaptive restart for accelerated gradient schemes, Found. Comput. Math., 15, 3, 715-732 (2015) · Zbl 1320.90061
[30] Passty, G. B., Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72, 2, 383-390 (1979) · Zbl 0428.47039
[31] Padcharoen, A.; Kitkuan, D.; Kumam, W.; Kumam, P., Tseng methods with inertial for solving inclusion problems and application to image deblurring and image recovery problems, Comput. Math. Methods (2020)
[32] Polyak, B. T., Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4, 1-17 (1964) · Zbl 0147.35301
[33] Raguet, H.; Fadili, J.; Peyré, G., A generalized forward-backward splitting, SIAM J. Imaging Sci., 6, 1199-1226 (2013) · Zbl 1296.47109
[34] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 877-898 (1976) · Zbl 0358.90053
[35] Sahu, D. R.; Yao, J. C.; Verma, M.; Shukla, K. K., Convergence rate analysis of proximal gradient methods with applications to composite minimization problems, Optimization, 70, 1, 75-100 (2021) · Zbl 1459.65075
[36] Suparatulatorn, R.; Khemphet, A., Tseng type methods for inclusion and fixed point problems with applications, Mathematics, 7, 12, 1175 (2019)
[37] Suparatulatorn, R.; Khemphet, A.; Charoensawan, P.; Suantai, S.; Phudolsitthiphat, N., Generalized self-adaptive algorithm for solving split common fixed point problem and its application to image restoration problem, Int. J. Comput. Math., 97, 7, 1431-1443 (2020) · Zbl 07476003
[38] Suparatulatorn, R.; Charoensawan, P.; Poochinapan, K.; Dangskul, S., An algorithm for the split feasible problem and image restoration, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 115, 1 (2021) · Zbl 07299282
[39] Suantai, S.; Kankam, K.; Cholamjiak, P.; Cholamjiak, W., A parallel monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with a graph applicable in signal recovery, Comput. Appl. Math. (2021) · Zbl 1483.47102
[40] Tseng, P., A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38, 431-446 (2000) · Zbl 0997.90062
[41] Takahashi, W.; Takeuchi, Y.; Kubota, R., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341, 1, 276-286 (2008) · Zbl 1134.47052
[42] Tan, B.; Zhou, Z.; Qin, X., Accelerated projection-based forward-backward splitting algorithms for monotone inclusion problems, J. Appl. Anal. Comput., 10, 5, 2184-2197 (2020) · Zbl 07331985
[43] Wang, Z.; Bovik, A. C.; Sheikh, H. R.; Simoncelli, E. P., Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13, 4, 600-612 (2004)
[44] Yambangwai, D.; Khan, S. A.; Dutta, H.; Cholamjiak, W., Image restoration by advanced parallel inertial forward-backward splitting methods, Soft Comput. (2021) · Zbl 1498.94019
[45] Zhao, T. Y.; Wang, D. Q.; Ceng, L. C.; He, L.; Wang, C. Y.; Fan, H. L., Quasi-inertial Tseng’s extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 42, 69-90 (2020) · Zbl 07336635
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