Browse > Article
http://dx.doi.org/10.22771/nfaa.2022.27.02.12

ON STRONG CONVERGENCE THEOREMS FOR A VISCOSITY-TYPE TSENG'S EXTRAGRADIENT METHODS SOLVING QUASIMONOTONE VARIATIONAL INEQUALITIES  

Wairojjana, Nopparat (Applied Mathematics Program, Faculty of Science and Technology Valaya Alongkorn Rajabhat University)
Pholasa, Nattawut (School of Science, University of Phayao)
Pakkaranang, Nuttapol (Mathematics and Computing Science Program, Faculty of Science and Technology Phetchabun Rajabhat University)
Publication Information
Nonlinear Functional Analysis and Applications / v.27, no.2, 2022 , pp. 381-403 More about this Journal
Abstract
The main goal of this research is to solve variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces numerically. The main advantage of these iterative schemes is the ease with which step size rules can be designed based on an operator explanation rather than the Lipschitz constant or another line search method. The proposed iterative schemes use a monotone and non-monotone step size strategy based on mapping (operator) knowledge as a replacement for the Lipschitz constant or another line search method. The strong convergences have been demonstrated to correspond well to the proposed methods and to settle certain control specification conditions. Finally, we propose some numerical experiments to assess the effectiveness and influence of iterative methods.
Keywords
Variational inequality problem; Tseng's extragradient method; strong convergence result; quasimonotone operator; Lipschitz continuity;
Citations & Related Records
Times Cited By KSCI : 8  (Citation Analysis)
연도 인용수 순위
1 L.C. Ceng, X. Qin, Y. Shehu and J.C. Yao, Mildly inertial subgradient extragradient method for variational inequalities involving an asymptotically nonexpansive and finitely many nonexpansive mappings, Mathematics, 7 (2019), Article ID 881.
2 C.M. Elliott, Variational and quasivariational inequalities: applications to free-boundary problems (Claudio Baiocchi and antonio Capelo), SIAM Rev., 29(2) (1987), 314-315.   DOI
3 H.A. Hammad, H. Rehman and M.D. la Sen, Advanced algorithms and common solutions to variational inequalities, Symmetry, 12 (2020), Article ID 1198.
4 A.N. Iusem and B.F. Svaiter, A variant of korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321.   DOI
5 G. Kassay, J. Kolumban and Z. Pales, On nash stationary points, Publ. Math. Debrecen, 54(3-4) (1999), 267-279.   DOI
6 G. Kassay, J. Kolumban and Z. Pales, Factorization of minty and stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377-389.   DOI
7 J.K. Kim and Salahuddin, Extragradient methodsfor generalized mixed equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Funct. Anal. Appl., 22(4) (2017), 693-709.   DOI
8 Z. Liu, S. Migorski and S. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equ., 263 (2017), 3989-4006.   DOI
9 D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Classic in Applied Mathematics, Society for Industrial and Applied Mathematics, (SIAM), Philadelphia, PA, 2000.
10 G. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonom. i Mat. Metody, 12(4) (1976), 747-756.
11 I. Konnov, Equilibrium models and variational inequalities, 210, Elsevier, 2007.
12 H. Rehman, P. Kumam, Y.J. Cho, Y.I. Suleiman and W. Kumam, Modified popov's explicit iterative algorithms for solving pseudomonotone equilibrium problems, Optim. Methods Softw., 36(1) (2021), 82-113.   DOI
13 A.S. Antipin, On a method for convex programs using a symmetrical modification of the Lagrange function, Economika i Matem. Metody., 12 (1976), 1164-1173.
14 L.C. Ceng, A. Petru,sel and J.C. Yao, On mann viscosity subgradient extragradient algorithms for fixed point problems of finitely many strict pseudocontractions and variational inequalities, Mathematics, 7 (2019), Article ID 925.
15 L.C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2019), 715-740.
16 Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335.   DOI
17 P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.   DOI
18 K. Muangchoo, A Viscosity type Projection Method for Solving Pseudomonotone Variational Inequalities, Nonlinear Funct. Anal. Appl., 26(2) (2021), 347-371.   DOI
19 Y. Censor, A. Gibali and S. Reich, Extensions of korpelevich extragradient method for the variational inequality problem in euclidean space, Optimization, 61 (2012), 1119-1132.   DOI
20 Z. Liu, S. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equ., 260 (2016), 6787-6799.   DOI
21 H. Rehman, A. Gibali, P. Kumam and K. Sitthithakerngkiet, Two new extragradient methods for solving equilibrium problems, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat., 115(2) (2021), Article ID 75.
22 H. Rehman, N. Pakkaranang, A. Hussain and N. Wairojjana, A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces, J. Math. Comput. Sci., 22(1) (2020), 38-48.   DOI
23 K. Muangchoo, H. Rehman and P. Kumam, Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications, J. Appl. Math. Comput. , 67(1-2) (2021), 891-917.   DOI
24 H. Rehman, P. Kumam, Q.L. Dong and Y.J. Cho, A modified self-adaptive extragradient method for pseudomonotone equilibrium problem in a real Hilbert space with applications, Math. Methods Appl. Sci., 44(5) (2020), 3527-3547.
25 H. Rehman, P. Kumam, A. Gibali and W. Kumam, Convergence analysis of a general inertial projection-type method for solving pseudomonotone equilibrium problems with applications, J. Inequal. Appl., 2021 (2021), Article ID 63.
26 N. Wairojjana and N. Pakkaranang, Halpern Tseng's extragradient methods for solving variational inequalities involving semistrictly quasimonotone operator, Nonlinear Funct. Anal. Appl., 27(1) (2022), 121-140.
27 H.K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65(1) (2002), 109-113.   DOI
28 M.A. Noor, Some iterative methods for nonconvex variational inequalities, Math. Comput. Modelling, 54 (11-12) (2011), 2955-2961.   DOI
29 H. Rehman, P. Kumam, I.K. Argyros, W. Deebani and W. Kumam, Inertial extragradient method for solving a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces with application in variational inequality problem, Symmetry, 12 (2020), Article ID 503.
30 G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.
31 P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38(2) (2000), 431-446.   DOI
32 A. Nagurney, Network economics: a variational inequality approach, Kluwer Academic Publishers Group, Dordrecht, 1993. https://doi.org/10.1007/978-94-011-2178-1   DOI
33 H. Rehman, P. Kumam, A.B. Abubakar and Y.J. Cho, The extragradient algorithm with inertial effects extended to equilibrium problems, Comput. Appl. Math., 39(2) (2020), Article ID 100.
34 A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.   DOI
35 H. Rehman, P. Kumam, Y.J. Cho and P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibrium problems, J. Inequal. Appl., 2019 (2019), Article ID 282.
36 H. Rehman, P. Kumam, W. Kumam, M. Shutaywi and W. Jirakitpuwapat, The inertial sub-gradient extra-gradient method for a class of pseudo-monotone equilibrium problems, Symmetry, 12(3) (2020), Article ID 463.
37 J. Yang and H. Liu, A modified projected gradient method for monotone variational inequalities, J. Optim. Theory Appl., 179 (1) (2018), 197-211.   DOI
38 L. Zhang, C. Fang and S. Chen, An inertial subgradient-type method for solving single-valued variational inequalities and fixed point problems, Numer. Algorithms, 79(3) (2018), 941-956.   DOI
39 H.H. Bauschke and P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, New York, 2011.
40 J. Yang, H. Liu and Z. Liu, Modified subgradient extragradient algorithms for solving monotone variational inequalities, Optimization, 67 (2018), 2247-2258.   DOI
41 P. Yordsorn, P. Kumam, H. Rehman and A.H. Ibrahim, A weak convergence self-adaptive method for solving pseudomonotone equilibrium problems in a real Hilbert space, Mathematics, 8 (2020), Article ID 1165.
42 T.Y. Zhao, D.Q. Wang, L.C. Ceng, L. He, C.Y. Wang and H.L. Fan, Quasi-inertial Tseng's extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 42(1) (2021), 69-90.   DOI