On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis
Volume 27, Issue 1, pp 42--58
http://dx.doi.org/10.22436/jmcs.027.01.04
Publication Date: February 10, 2022
Submission Date: November 29, 2021
Revision Date: December 17, 2021
Accteptance Date: January 06, 2022
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Authors
N. Wairojjana
- Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage, 1 Moo 20 Phaholyothin Road, Klong Neung, Klong Luang, Pathumthani 13180, Thailand.
N. Pakkaranang
- Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand.
S. Noinakorn
- Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand.
Abstract
The main objective of this research is to find the numerical solution of variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces. The main advantage of these iterative schemes is that they allow the uncomplicated calculation of step size rules that depend on the knowledge of an operator explanation instead of the Lipschitz constant or some other line search method. The proposed iterative schemes follow a monotone and non-monotone step size procedure based on mapping (operator) information as a replacement for its Lipschitz constant or some other line search method. The strong convergences are well proven, analogous to the proposed methods, and impose certain control specification conditions. Finally, to verify the effectiveness of the iterative methods, we present some numerical experiments.
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ISRP Style
N. Wairojjana, N. Pakkaranang, S. Noinakorn, On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis, Journal of Mathematics and Computer Science, 27 (2022), no. 1, 42--58
AMA Style
Wairojjana N., Pakkaranang N., Noinakorn S., On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis. J Math Comput SCI-JM. (2022); 27(1):42--58
Chicago/Turabian Style
Wairojjana, N., Pakkaranang, N., Noinakorn, S.. "On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis." Journal of Mathematics and Computer Science, 27, no. 1 (2022): 42--58
Keywords
- Variational inequality problem
- Tseng's extragradient method
- strong convergence theorems
- quasimonotone operator
- Lipschitz continuity
MSC
References
-
[1]
A. S. Antipin, On a method for convex programs using a symmetrical modification of the lagrange function, Ekonomika i Matem. Metody, 12 (1976), 1164--1173
-
[2]
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2017)
-
[3]
L. C. Ceng, Asymptotic inertial subgradient extragradient approach for pseudomonotone variational inequalities with fixed point constraints of asymptotically nonexpansive mappings, Commun. Optim. Theory, 2020 (2020), 21 pages
-
[4]
L. C. Ceng, A. Petrusel, X. Qin, J. C. Yao, A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93--108
-
[5]
L. C. Ceng, A. Petrusel, C. F. Wen, J. C. Yao, Inertial-like subgradient extragradient methods for variational inequalities and fixed points of asymptotically nonexpansive and strictly pseudocontractive mappings, Mathematics, 7 (2019), 19 pages
-
[6]
L. C. Ceng, A. Petrusel, J. C. Yao, On Mann viscosity subgradient extragradient algorithms for fixed point problems of finitely many strict pseudocontractions and variational inequalities, Mathematics, 7 (2019), 14 pages
-
[7]
L. C. Ceng, X. Qin, Y. Shehu, J. C. Yao, Mildly inertial subgradient extragradient method for variational inequalities involving an asymptotically nonexpansive and finitely many nonexpansive mappings, Mathematics, 7 (2019), 19 pages
-
[8]
L.-C. Ceng, M.-J. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2019), 715--740
-
[9]
L. C. Ceng, M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., 2020 (2020), 19 pages
-
[10]
Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2010), 318--335
-
[11]
Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119--1132
-
[12]
J. F. Chu, F. Gharehgazlouei, S. Heidarkhani, A. Solimaninia, Three nontrivial solutions for Kirchhoff--type variational--hemivariational inequalities, Results Math., 68 (2015), 71--91
-
[13]
C. M. Elliott, Variational and quasivariational inequalities applications to free--boundary problems (Claudio baiocchi and Antonio Capelo), SIAM Rev., 29 (1987), 314--315
-
[14]
A. N. Iusem, B. F. Svaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309--321
-
[15]
G. Kassay, J. Kolumban, Z. Pales, On nash stationary points, Publ. Math. Debrecen, 54 (1999), 267--279
-
[16]
G. Kassay, J. Kolumban, Z. Pales, Factorization of minty and stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377--389
-
[17]
D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, SIAM, Philadelphia (2000)
-
[18]
I. Konnov, Equilibrium models and variational inequalities, Elsevier B. V., Amsterdam (2007)
-
[19]
G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747--756
-
[20]
L. Liu, S. Y. Cho, J. C. Yao, Convergence analysis of an inertial Tseng's extragradient algorithm for solving pseudomonotone variational inequalities and applications, J. Nonlinear Var. Anal., 5 (2021), 627--644
-
[21]
P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899--912
-
[22]
A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241 (2000), 46--55
-
[23]
K. Muangchoo, H. ur Rehman, P. Kumam, Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications, J. Appl. Math. Comput., 67 (2021), 891--917
-
[24]
M. A. Noor, Some iterative methods for nonconvex variational inequalities, Comput. Math. Model., 21 (2010), 97--108
-
[25]
F. U. Ogbuisi, O. S. Iyiola, J. Ngnotchouye, T. Shumba, On inertial type self-adaptive iterative algorithms for solving pseudomonotone equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2021 (2021), 18 pages
-
[26]
G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413--4416
-
[27]
D. Q. Tran, M. L. Dung, V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization, 749--766 (2008),
-
[28]
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431--446
-
[29]
H. ur Rehman, A. Gibali, P. Kumam, K. Sitthithakerngkiet, Two new extragradient methods for solving equilibrium problems, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 115 (2021), 25 pages
-
[30]
H. ur Rehman, P. Kumam, A. B. Abubakar, Y. J. Cho, The extragradient algorithm with inertial effects extended to equilibrium problems, Comput. Appl. Math., 39 (2020), 26 pages
-
[31]
H. ur Rehman, P. Kumam, Y. J. Cho, P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibirum problems, J. Inequal. Appl., 1 (2019), 25 pages
-
[32]
H. ur Rehman, P. Kumam, Q.-L. Dong, 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 (2020), 3527--3547
-
[33]
H. ur Rehman, P. Kumam, A. Gibali, W. Kumam, Convergence analysis of a general inertial projection-type method for solving pseudomonotone equilibrium problems with applications, J. Inequal. Appl., 2021 (2021), 27 pages
-
[34]
H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109--113
-
[35]
J. Yang, H. Liu, A modified projected gradient method for monotone variational inequalities, J. Optim. Theory Appl., 179 (2018), 197--211
-
[36]
J. Yang, H. W. Liu, Z. X. Liu, Modified subgradient extragradient algorithms for solving monotone variational inequalities, Optimization, 67 (2018), 2247--2258
-
[37]
P. Yordsorn, P. Kumam, H. ur Rehman, Modified two-step extragradient method for solving the pseudomonotone equilibrium programming in a real Hilbert space, Carpathian J. Math., 36 (2020), 313--330
-
[38]
P. Yordsorn, P. Kumam, H. ur Rehman, A. H. Ibrahim, A weak convergence self-adaptive method for solving pseudomonotone equilibrium problems in a real Hilbert space, Mathematics, 8 (2020), 24 pages
-
[39]
L. X. Zhang, C. J. Fang, S. L. Chen, An inertial subgradient-type method for solving single-valued variational inequalities and fixed point problems, Numer. Algorithms, 79 (2018), 941--956
-
[40]
T.-Y. Zhao, D.-Q. Wang, L.-C. Ceng, L. He, C.-Y. Wang, 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 (2021), 69--90