An inexact operator splitting method for general mixed variational inequalities

Abstract

The present paper aims to deal with an inexact implicit method with a variable parameter for general
mixed variational inequalities in the setting of real Hilbert spaces. Under standard assumptions, the global
convergence of the proposed method is proved. Numerical example is presented to illustrate the proposed
method and convergence result. The results and method presented in this paper generalize, extend and unify
some known results in the literature.

Keywords

• General mixed variational inequalities
• implicit methods
• resolvent operator
• inexact methods
• self-adaptive rules

References

1. [1] A. Bnouhachem, A self-adaptive method for solving general mixed variational inequalities, J. Math. Anal. Appl. 309 (2005), 136-150.
2. [2] A. Bnouhachem, M.A. Noor and Th.M. Rassias, Three-step iterative algorithm for mixed variational inequalities, Appl. Math. and Comput. 183(1) (2006), 436-446.
3. [3] A. Bnouhachem and M.A. Noor, Inexact proximal point method for general variational inequalities, J. Math. Anal. Appl. 324(2) (2006), 1195-1212.
4. [4] A. Bnouhachem, An inexact implicit method for general mixed variatioanl inequalities, J. Comput. Appl. Math. 200 (2007), 377-387.
5. [5] A. Bnouhachem, M.A. Noor, Numerical methods for general mixed variational inequalities, App. Math. Comput. 204 ( 2008), 27-36.
6. [6] H. Brezis, Operateurs maximaux monotone et semigroupes de contractions dans les espace d'Hilbert, North-Holland, Amsterdam, Holland, 1973.
7. [7] W. Cholmajiak P. Kitisak and D. Yambangwai, An inertial parallel CQ subgradient extragradient method for variational inequalities application to signal-image recovery, Results in Nonlinear Analysis, 4(4) (2021), 217-234.
8. [8] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Springer, Berlin 2003.

9. [9] M. Farid, W. Cholamjiak, R. Ali, K.R. Kazmi, A new shrinking projection algorithm for a generalized mixed variational-like inequality problem and asymptotically quasi-φ-nonexpansive mapping in a Banach space, Revista de la Real Academia de Ciencias Exactas, F�sicas y Naturales. Serie A. Matemáticas, 115(3) (2021), 1-28.
10. [10] F. Giannessi, A. Maugeri and P.M. Pardalos, (eds.), Equilibrium Problems and Variational Models, Kluwer Academic, Dordrecht (2001).
11. [11] Z. Ge, G. Qian and D.R. Han, Global convergence of an nexact operator splitting method for monotone variational inequalities, J. Indus. Manag. Optim. 7(4) (2011), 1013-1026.
12. [12] R. Glowinski, J.L. Lions and R. Tremoliers, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981.
13. [13] B.S. He, Inexact implicit methods for monotone general variational inequalities, Math. Program. 86 (1999), 199-217.
14. [14] B.S. He, H. Yang, Q. Meng and D.R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities, J. Optim. Theory Appl. 112(1) (2002), 129-143.
15. [15] B.S. He, L.Z. Liao and S.L. Wang, Self-adaptive operator splitting methods for monotone variational inequalities, Numer. Math. 94 (2003), 715-737.
16. [16] T. Kai and X. Fuquan, A projection type algorithm for solving generalized mixed variational inequalities, Act. Math. Scien. 36B(6) (2016), 1619-1630.
17. [17] J.L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Apl. Math. 20 (1967), 493-512.
18. [18] M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities, J. Global Optim. 41(3) (2008), 417-426.
19. [19] M. Li and X.M. Yuan, An improved Goldstein's type method for a class of variant variational inequalities, J. Comput. Appl. Math. 214(1) (2008), 304-312.
20. [20] M.A. Noor, An implicit method for mixed variational inequalities, Appl. Math. Lett. 11 (1998), 109-113.
21. [21] M.A. Noor, Pseudomonotone general mixed variational inequalities, Appl. Math. Comput. 141 (2003), 529-540.
22. [22] M.A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004), 199-277.
23. [23] S. Suanta, P. Peeyad, D. Yambangwai and W. Cholamjiak, A parallel-viscosity-type subgradient extragradient-line method for finding the common solution of variational inequality problems applied to image restoration problems, Mathematics, 8(2) (2020), 248.
24. [24] S. Ullah and M.A. Noor, An efficient method solving new general mixed variational inequalities, J. Inequal. Special Funct. 11(3) (2020), 1-9.
25. [25] S.L. Wang, H. Yang and B.S. He, Inexact implicit method with variable parameter for mixed monotone variational in- equalities, J. Optim. Theory Appl. 111(2) (2001), 431-443.
26. [26] L.C. Zeng and J.C. Yao, Convergence analysis of a modified inexact implicit method for general mixed monotone variational inequalities, Math. Meth. Oper. Res. 62 (2005), 211-224.