A modified parallel monotone hybrid algorithm for a finite family of $\mathcal{G}$-nonexpansive mappings apply to a novel signal recovery

Abstract

In this work, we investigate the strong convergence of the sequences generated by the shrinking projection method and the parallel monotone hybrid method to find a common fixed point of a finite family of $\mathcal{G}$-nonexpansive mappings under suitable conditions in Hilbert spaces endowed with graphs. We also give some numerical examples and provide application to signal recovery under situation without knowing the type of noises. Moreover, numerical experiments of our algorithms which are defined by different types of blurred matrices and noises on the algorithm to show the efficiency and the implementation for LASSO problem in signal recovery.

Keywords

• Shrinking projection method
• $G-$nonexpasive mapping
• Common fixed point
• Hilbert space
• Signal recovery

References

1. [1] S.M.A. Aleomraninejad, S. Rezapour, N. Shahzad, Some fixed point results on a metric space with a graph. Topology and its Applications. 159(3) 2012 659-663.
2. [2] M.R. Alfuraidan, On monotone Ciric quasi-contraction mappings with a graph. Fixed Point Theory and Applications, 2015(1) (2015) 93.
3. [3] M.R. Alfuraidan, M.A. Khamsi, Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory and Applications, 2015(1) (2015) 44.
4. [4] P. K. Anh, D.V. Hieu, Parallel and sequential hybrid methods for a finite family of asymptotically quasi ϕ-nonexpansive mappings. Journal of Applied Mathematics and Computing, 48(1) (2015) 241-263.
5. [5] P. K. Anh, D.V. Hieu, Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems. Vietnam Journal of Mathematics, 44(2) (2016) 351-374.
6. [6] C.E. Chidume, J.N. Ezeora, Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-contractive mappings. Fixed Point Theory and Applications, 2014(1) (2014)111.
7. [7] P. Cholamjiak, A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces. Numerical Algorithms, 71(4) (2016) 915-932.
8. [8] W. Cholamjiak, P. Cholamjiak, S. Suantai, An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. Journal of Fixed Point Theory and Applications, 20(1) (2018) 42.

9. [9] P. Cholamjiak, S. Suantai, P. Sunthrayuth, An explicit parallel algorithm for solving variational inclusion problem and fixed point problem in Banach spaces. Banach Journal of Mathematical Analysis, 14(1) (2020) 20-40.
10. [10] Q. Dong, D. Jiang, P. Cholamjiak, Y. Shehu, A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions. Journal of Fixed Point Theory and Applications, 19(4) (2017) 3097-3118.
11. [11] J.C. Dunn, Convexity, monotonicity, and gradient processes in Hilbert space. Journal of Mathematical Analysis and Appli- cations, 53(1) (1976) 145-158.
12. [12] A.G. Gebrie, R. Wangkeeree, Parallel proximal method of solving split system of fixed point set constraint minimization problems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 114(1) (2020) 1-29.
13. [13] O. Guler, On the convergence of the proximal point algorithm for convex minimization. SIAM Journal on Control and Optimization, 29(2) (1991) 403-419.
14. [14] D.V. Hieu, Parallel and cyclic hybrid subgradient extragradient methods for variational inequalities. Afrika Matematika, 28(5) (2017) 677-692.
15. [15] J. Jachymski, The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathe- matical Society, 136(4) (2008) 1359-1373.
16. [16] S.A. Khan, S. Suantai, W. Cholamjiak, Shrinking projection methods involving inertial forward-backward splitting methods for inclusion problems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113(2) (2019) 645-656.
17. [17] S.A. Khan, W. Cholamjiak, K.R. Kazmi, An inertial forward-backward splitting method for solving combination of equi- librium problems and inclusion problems. Computational and Applied Mathematics, 37(5) (2018) 6283-6307.
18. [18] K. Kankam, N. Pholasa, P. Cholamjiak, On convergence and complexity of the modified forward-backward method involv- ing new linesearches for convex minimization. Mathematical Methods in the Applied Sciences, 42(5) (2019) 1352-1362.
19. [19] K. Kankam, P. Cholamjiak, Strong convergence of the forward–backward splitting algorithms via linesearches in Hilbert spaces. Applicable Analysis, 1-20 (2021).
20. [20] K. Kankam, N. Pholasa, P. Cholamjiak, Hybrid forward-backward algorithms using linesearch rule for minimization prob- lem. Thai Journal of Mathematics, 17(3) (2019) 607-625.
21. [21] B. Martinet, Brvèv communication. Régularisation d’inéquations variationnelles par approximations successives. Revue franaise d’informatique et de recherche opérationnelle. Série rouge, 4(R3) (1907) 154-158.
22. [22] C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods and Applications, 64(11) (2006) 2400-2411.
23. [23] T. Minh Tuyen, R. Promkam, P. Sunthrayuth, Strong convergence of a generalized forward-backward splitting method in reflexive Banach spaces. Optimization, 71(6) (2022) 1483-1508.
24. [24] K. Nakajo, W. Takahashi,Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications, 279(2) (2003) 372-379.
25. [25] N. Parikh, S. Boyd, Proximal algorithms. Foundations and Trends in Optimization, 1(3) (2014) 127-239.
26. [26] N. Pholasaa, P. Cholamjiaka, Y. J. Cho, Modified forward-backward splitting methods for accretive operators in Banach spaces. Journal of Nonlinear Sciences and Applications, 9 (2016) 2766-2778.
27. [27] S. Reich, T.M. Tuyen, N. M. Trang, Parallel iterative methods for solving the split common fixed point problem in Hilbert spaces. Numerical Functional Analysis and Optimization, 41(7) (2020) 778-805.
28. [28] R.T. Rockafellar, Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14(5) (1976) 877-898.
29. [29] P. Sridarat, R. Suparaturatorn, S. Suantai, Y. J. Cho, Convergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs. Bulletin of the Malaysian Mathematical Sciences Society, 42(5) (2019) 2361-2380.
30. [30] K. Sitthithakerngkiet, J. Deepho, P. Kumam, Modified hybrid steepest method for the split feasibility problem in image recovery of inverse problems. Numerical Functional Analysis and Optimization, 38(4) (2017)507-522.
31. [31] S. Suantai, P. Jailoka, A. Hanjing, An accelerated viscosity forward-backwardsplitting algorithm with the linesearch process for convex minimization problems. Journal of Inequalities and Applications, 2021(1) (2021) 1-19.
32. [32] S. Suantai, K. Kankam, P. Cholamjiak, A novel forward-backward algorithm for solving convex minimization problem in Hilbert spaces. Mathematics, 8(1) (2020) 42.
33. [33] S. Suantai, K. Kankam, P. Cholamjiak, A projected forward-backward algorithm for constrained minimization with appli- cations to image inpainting. mathematics, 9(8) (2021) 890.
34. [34] S. Suantai, M.A. Noor, K. Kankam, P. Cholamjiak, Novel forward–backward algorithms for optimization and applications to compressive sensing and image inpainting. Advances in Difference Equations, 2021(1) (2021) 1-22.
35. [35] S. Suantai, M. Donganont, W. Cholamjiak, Hybrid methods for a countable family of G-nonexpansive mappings in Hilbert spaces endowed with graphs. Mathematics, 7(10) (2019) 936.
36. [36] S. Suantai, P. Cholamjiak, P. Sunthrayuth, Iterative methods with perturbations for the sum of two accretive operators in q-uniformly smooth Banach spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113(1) (2019) 203-223.
37. [37] R. Suparatulatorn, S. Suantai, W. Cholamjiak, Hybrid methods for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs. AKCE International Journal of Graphs and Combinatorics, 14(2) (2017) 101-111.
38. [38] A. Taiwo, L.O. Jolaoso, O.T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split com- mon fixed point problems. Bulletin of the Malaysian Mathematical Sciences Society, 43(2) (2022) 1893-1918.
39. [39] J.Tiammee, A.Kaewkhao, S.Suantai, OnBrowdersconvergencetheoremandHalperniterationprocessforG-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory and Applications, 2015(1) (2015) 187.
40. [40] O. Tripak, Common fixed points of G-nonexpansive mappings on Banach spaces with a graph. Fixed Point Theory and Applications, 2016(1) (2016) 87.
41. [41] C. Wang, N. Xiu, Convergence of the gradient projection method for generalized convex minimization. Computational Optimization and Applications, 16(2) (2000) 111-120.
42. [42] H.K. Xu, Averaged mappings and the gradient-projection algorithm. Journal of Optimization Theory and Applications, 150(2) (2011) 360-378.
43. [43] D. Yambangwai, S. Aunruean, T. Thianwan, A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph. Numerical Algorithms, 84(2) (2020) 537-565. ISO 690
44. [44] J. Yang, P. Cholamjiak, P. Sunthrayuth, Modified Tseng’s splitting algorithms for the sum of two monotone operators in Banach spaces. AIMS Math, 6(5) (2021) 4873-4900.