An inertial parallel CQ subgradient extragradient method for variational inequalities application to signal-image recovery

Abstract

In this paper, we introduce an inertial parallel CQ subgradient extragradient method for finding a common solutions of variational inequality problems. The novelty of this paper is using linesearch methods to find unknown L constant of L-Lipschitz continuous mappings. Strong convergence theorem has been proved under some suitable conditions in Hilbert spaces. Finally, we show applications to signal and image recovery, and show the good efficiency of our proposed algorithm when the number of subproblems is increasing.

Keywords

• CQ algorith
• Subgradient extragradient method
• Parallel algorithm
• Signal recovery
• Image restoration

Kaynakça

1. [1] Alber, Y., Ryazantseva, I.: Nonlinear ill-posed problems of monotone type. Springer. London,(2006).
2. [2] Assad, N., Kirk, W.: Fixed point theorems for set-valued mappings of contractive type. Pacific Journal of Mathematics. 43(3), 553-562, (1972).
3. [3] Anh, P. K., Van Hieu, D.: Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems. Vietnam Journal of Mathematics. 44(2), 351-374, (2016).
4. [4] Anh, P. K., Van Hieu, D.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi `-nonexpansive mappings. Journal of Applied Mathematics and Computing. 48(1-2), 241-263, (2015).
5. [5] Bauschke, H. H., Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM review. 38(3), 367-426, (1996).
6. [6] Censor, Y., Chen, W., Combettes, P. L., Davidi, R., Herman, G. T.: On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Computational Optimization and Applications, 51(3), 1065-1088, (2012).
7. [7] Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optimization Methods and Software, 26(4-5), 827-845, (2011).
8. [8] Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. Journal of Optimization Theory and Applications. 148(2), 318-335, (2011).

9. [9] Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set-Valued and Variational Analysis, 20(2), 229-247, (2012).
10. [10] Deepho, J., Kumam, W., Kumam, P.: A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems. Journal of Mathematical Modeling and Algorithms in Operations Research, 13(4), 405-423, (2014).
11. [11] Fang, C., Chen, S.: Some extragradient algorithms for variational inequalities. Advances in variational and hemivariational inequalities, 145-171, Adv. Mech Math., vol. 33. Springer, Cham (2015).
12. [12] Engl, H. W., Hanke, M., Neubauer, A. Regularization of Inverse Problems., Dordrecht: Kluwer Academic Publishers, 2000.
13. [13] Eslamian, M.: Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems. Optimization Letters, 7(3), 547-557, (2013).
14. [14] Gibali, A.: A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces. Journal of Nonlinear Analysis and Optimization: Theory and Applications, 6(1), 41-5, (2015).
15. [15] Glowinski, R., Tremolieres, R., Lions, J. L.: Numerical analysis of variational inequalities. Elsevier, (2011).
16. [16] Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Mathematica, 115(1), 271-310, (1966).
17. [17] Van Hieu, D., Anh, P. K.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Computational Optimization and Applications, 66(1), 75-96, (2017).
18. [18] Van Hieu, D.: Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. Journal of Applied Mathematics and Computing, 53(1-2), 531-554, (2017).
19. [19] Vogel, C.R. Computational Methods for Inverse Problems. PA: SIAM Philadelphia, 2002.
20. [20] Jiang, L., Su, Y.: Weak convergence theorems for equilibrium problems and nonexpansive mappings and nonspreading mappings in Hilbert spaces. Commun. Korean Math. Soc, 27(3), 505-512, (2012).
21. [21] Jitpeera, T., Inchan, I., Kumam, P.: A general iterative algorithm combining viscosity method with parallel method for mixed equilibrium problems for a family of strict pseudo-contractions, Journal of Applied Mathematics and Informatics, 29(34), 621-639, (2011).
22. [22] Jung, J. S.: Convergence of approximating fixed pints for multivalued nonself-mappings in Banach spaces. Korean J. Math, 16(2), 215-231, (2008).
23. [23] Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonlinear Analysis: Theory, Methods and Applications, 61(1-2), 51-60, (2005).
24. [24] Komal, S., Kumam, P.: A Modified Subgradient Extragradient Algorithm with Inertial Effects. Communications in Mathematics and Applications, 10(2), 267-280, (2019).
25. [25] Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody, (1976).
26. [26] Liu, H. B.: Convergence theorems for a finite family of nonspreading and nonexpansive multivalued mappings and equilibrium problems with application. Math. Appl, 3, 49-61, (2013).
27. [27] Martinez-Yanes, C., Xu, H. K.: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods and Applications, 64(11), 2400-2411, (2006).
28. [28] Pietramala, P.: Convergence of approximating fixed points sets for multivalued nonexpansive mappings. Commentationes Mathematicae Universitatis Carolinae, 32(4), 697-701, (1991).
29. [29] Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society, 149(1), 75-88, (1970).
30. [30] Shahzad, N., Zegeye, H.: Strong convergence results for nonself multimaps in Banach spaces. Proceedings of the American Mathematical Society, 136(2), 539-548, (2008).
31. [31] Shahzad, N., Zegeye, H.: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Analysis: Theory, Methods and Applications, 71(3-4), 838-844, (2009).
32. [32] Song, Y., Wang, H.: Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal. 70 1547-1556, (2009).
33. [33] Shehu, Y., Iyiola, O.S.: Strong convergence result for monotone variational inequalities. Numerical Algorithms, (2016).
34. [34] Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765-776, (1999).
35. [35] Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in Hilbert space. Math. Progr. 87, 189-202, (2000).
36. [36] Stark, H.: Image Recovery Theory and Applications. Academic, Orlando, (1987).
37. [37] Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, (2000).
38. [38] Van Hieu, D.: Parallel and cyclic hybrid subgradient extragradient methods for variational inequalities. Afrika Matematika, 28(5-6), 677-692, (2017).
39. [39] Wang, Z.M., Cho, S.Y., Su, Y.: Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems. Thai Journal of Mathematics, 15, 835-8600, (2017).
40. [40] Yao, Y., Liou, Y.C.: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Probl. 24, Article ID 015015. (2008).
41. [41] Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpensive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473-504. Elsevier, Amsterdam, (2001).
42. [42] Yimer, S. E., Kumam, P., Gebrie A. G., and Wangkeeree, R.: Inertial Method for Bilevel Variational Inequality Problems with Fixed Point and Minimizer Point Constraints. Mathematics. 7, 841, (2019).
43. [43] Zhao, J., Yang, Q.: Self-adaptive projection methods for the multiple-sets split feasibility problem. Inverse Problems, 27(3), 035009 (2011).