Modified subgradient extragradient method to solve variational inequalities
Authors
Kanikar Muangchoo
- Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381 Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand.
Abstract
In this paper, we introduce a new method to solve pseudomonotone variational inequalities with the Lipschitz condition in a real Hilbert space. This problem is a general mathematical problem in the sense that it unifies a number of the mathematical problems as a particular case, such as the optimization problems, the equilibrium problems, the fixed point problems, the saddle point problems and Nash equilibrium point problems. The new method is constructed around two methods: the extragradient method and the inertial method. The proposed method uses a new stepsize rule based on local operator information rather than its Lipschitz constant or any other line search method. The proposed method does not require any knowledge of the Lipschitz constant of an operator. The strong convergence of the proposed method is well-established. Finally, we conduct a number of numerical experiments to determine the performance and superiority of the proposed method.
Share and Cite
ISRP Style
Kanikar Muangchoo, Modified subgradient extragradient method to solve variational inequalities, Journal of Mathematics and Computer Science, 25 (2022), no. 2, 133--149
AMA Style
Muangchoo Kanikar, Modified subgradient extragradient method to solve variational inequalities. J Math Comput SCI-JM. (2022); 25(2):133--149
Chicago/Turabian Style
Muangchoo, Kanikar. "Modified subgradient extragradient method to solve variational inequalities." Journal of Mathematics and Computer Science, 25, no. 2 (2022): 133--149
Keywords
- Variational inequality problem
- subgradient extragradient-like method
- strong convergence result
- Lipschitz continuity
- pseudomonotone mapping
MSC
- 65Y05
- 65K15
- 68W10
- 47H05
- 47H10
References
-
[1]
P. N. Anh, H. T. C. Thach, J. K. Kim, Proximal-like subgradient methods for solving multi-valued variational inequalities, Nonlinear Funct. Anal. Appl., 25 (2020), 437--451
-
[2]
A. S. Antipin, On a convex programming method using a symmetric modification of the Lagrange function, Econ. Math. Methods, 12 (1976), 1164--1173
-
[3]
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Cham (2017)
-
[4]
Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318--335
-
[5]
C. M. Elliott, Variational and quasivariational inequalities applications to free—boundary ProbLems. (claudio baiocchi and ant´onio capelo), SIAM Rev., 29 (1987), 314--315
-
[6]
P. T. Harker, J.-S. Pang, A damped-Newton method for the linear complementarity problem, Amer. Math. Soc., Providence, RI, 26 (1990), 265--284
-
[7]
X. Hu, J. Wang, Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487--1499
-
[8]
G. Kassay, J. Kolumban, Z. Pales, On Nash stationary points, Publ. Math. Debrecen, 54 (1999), 267--279
-
[9]
G. Kassay, J. Kolumban, Z. P ´ ales, Factorization of Minty and Stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377--389
-
[10]
J. K. Kim, A. H. Dar, Salahuddin, Existence theorems for the generalized relaxed pseudomonotone variational inequalities, Nonlinear Funct. Anal. Appl., 25 (2020), 25--34
-
[11]
D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York-London (1980)
-
[12]
I. V. Konnov, Equilibrium models and variational inequalities, Elsevier B. V., Amsterdam (2007)
-
[13]
G. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747--756
-
[14]
P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 899--912 (2008),
-
[15]
K. Muangchoo, H. U. Rehman, P. Kumam, Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications, J. Appl. Math. Comput., 2021 (2021), 1--27
-
[16]
A. Nagurney, Network economics: a variational inequality approach, Kluwer Academic Publishers Group, Dordrecht (1993)
-
[17]
M. A. Noor, M. Akhter, K. I. Noor, Inertial proximal method for mixed quasi variational inequalities, Nonlinear Funct. Anal. Appl.,, 8 (2003), 489--496
-
[18]
B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4 (1964), 1--17
-
[19]
H. U. Rehman, A. Gibali, P. Kumam, K. Sitthithakerngkiet, Two new extragradient methods for solving equilibrium problems, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM, 115 (2021), 1--25
-
[20]
H. U. Rehman, P. Kumam, A. B. Abubakar, Y. J. Cho, The extragradient algorithm with inertial effects extended to equilibrium problems, Comput. Appl. Math., 39 (2020), 1--26
-
[21]
H. U. Rehman, P. Kumam, I. K. Argyros, N. A. Alreshidi, Modified proximal-like extragradient methods for two classes of equilibrium problems in Hilbert spaces with applications, Comput. Appl. Math.,, 40 (2021), 1--26
-
[22]
H. U. Rehman, P. Kumam, Y. J. Cho, Y. I. Suleiman, W. Kumam, Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems, Optim. Methods Softw., 36 (2021), 82--113
-
[23]
H. U. Rehman, P. Kumam, Y. J. Cho, P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibrium problems, J. Inequal. Appl., 2019 (2019), 1--25
-
[24]
H. U. 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 (2021), 3527--3547
-
[25]
H. U. Rehman, P. Kumam, Q.-L. Dong, Y. Peng, W. Deebani, A new popov’s subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space, Optimization, 2020 (2020), 1--36
-
[26]
H. U. Rehman, P. Kumam, M. Shutaywi, N. A. Alreshidi, W. Kumam, Inertial optimization based two-step methods for solving equilibrium problems with applications in variational inequality problems and growth control equilibrium models, Energies, 13 (2022), 1--28
-
[27]
H. U. Rehman, P. Kumam, K. Sitthithakerngkiet, Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications, AIMS Math., 6 (2021), 1538--1560
-
[28]
H. U. Rehman, W. Kumam, P. Kumam, M. Shutaywi, A new weak convergence non-monotonic self-adaptive iterative scheme for solving equilibrium problems, AIMS Math., 6 (2021), 5612--5638
-
[29]
G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413--4416
-
[30]
W. Takahashi, Nonlinear functional analysis, Yokohama Publishers, Yokohama (2002)
-
[31]
H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109--113
-
[32]
J. Yang, Self-adaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities, Appl. Anal., 100 (2021), 1067--1078
-
[33]
J. Yang, H. Liu, Z. Liu, Modified subgradient extragradient algorithms for solving monotone variational inequalities, Optimization, 67 (2018), 2247--2258