Abstract
In this paper, we introduce a novel iterative method for finding the minimum-norm solution to a pseudomonotone variational inequality problem in Hilbert spaces. We establish strong convergence of the proposed method and its linear convergence under some suitable assumptions. Some numerical experiments are given to illustrate the performance of our method. Our result improves and extends some existing results in the literature.
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References
Anh PK, Thong DV, Vinh NT (2020) Improved inertial extragradient methods for solving pseudo-monotone variational inequalities. Optimization. https://doi.org/10.1080/02331934.2020.1808644
Censor Y, Gibali A, Reich S (2011a) The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl 148:318–335
Censor Y, Gibali A, Reich S (2011b) Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw 26:827–845
Censor Y, Gibali A, Reich S (2012a) Algorithms for the split variational inequality problem. Numer Algor 59:301–323
Censor Y, Gibali A, Reich S (2012b) Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61:1119–1132
Cottle RW, Yao JC (1992) Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl 75:281–295
Denisov SV, Semenov VV, Chabak LM (2015) Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern Syst Anal 51:757–765
Facchinei F, Pang JS (2003) Finite-dimensional variational inequalities and complementarity problems. Springer Series in Operations Research, vol I. Springer, New York
Gibali A, Reich S, Zalas R (2017) Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66:417–437
Harker PT (1984) A variational inequality approach for the determination of oligopolistic market equilibrium. Math Program 30:105–111
Kassay G, Reich S, Sabach S (2011) Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J Optim 21:1319–1344
Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, New York
Konnov IV (2001) Combined relaxation methods for variational inequalities. Springer-Verlag, Berlin
Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12:747–756
Kraikaew R, Saejung S (2014) Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl 163:399–412
Liu H, Yang J (2020) Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput Optim Appl 77:491–508
Murphy FH, Sherali HD, Soyster AL (1982) A mathematical programming approach for determining oligopolistic market equilibrium. Math Program 24:92–106
Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York
Reich S, Thong DV, Dong QL et al (2021) New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings. Numer Algor 87:527–549
Reich S, Thong DV, Cholamjiak P et al (2021) Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space. Numer Algor 88:813–835
Reich S, Shafrir I (1987) The asymptotic behavior of firmly nonexpansive mappings. Proc Amer Math Soc 101:246–250
Saejung S, Yotkaew P (2012) Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal 75:742–750
Shehu S, Iyiola OS, Yao JC (2021) New projection methods with inertial steps for variational inequalities. Optimization. https://doi.org/10.1080/02331934.2021.1964079
Shehu Y, Iyiola OS, Reich S (2021) A modified inertial subgradient extragradient method for solving variational inequalities. Optim Eng. https://doi.org/10.1007/s11081-020-09593-w
Thong DV, Hieu DV (2018) Weak and strong convergence theorems for variational inequality problems. Numer Algor 78:1045–1060
Thong DV, Vuong PT (2021) Improved subgradient extragradient methods for solving pseudomonotone variational inequalities in Hilbert spaces. Appl Numer Math 163:221–238
Vuong PT (2018) On the weak convergence of the extragradient method for solving pseudomonotone variational inequalities. J Optim Theory Appl 176:399–409
Yang J, Liu H, Liu Z (2018) Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization 67:2247–2258
Yang J (2021) Self-adaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities. Appl Anal 100:1067–1078
Acknowledgements
The authors are thankful to the handling editor and two anonymous reviewers for comments and remarks which substantially improved the quality of the paper. We also would like to express our gratitude to Professor Terry Friesz, Editor-in-Chief, for giving us the opportunity to revise and resubmit this manuscript.
Funding
The second and the third authors were partially supported by Vietnam Institute for Advanced Study in Mathematics (VIASM).
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Thong, D.V., Anh, P.K., Dung, V.T. et al. A Novel Method for Finding Minimum-norm Solutions to Pseudomonotone Variational Inequalities. Netw Spat Econ 23, 39–64 (2023). https://doi.org/10.1007/s11067-022-09569-6
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DOI: https://doi.org/10.1007/s11067-022-09569-6
Keywords
- Subgradient extragradient method
- Variational inequality problem
- Pseudomonotone operator
- Strong convergence
- Convergence rate