Abstract
In this paper, we present a regularized dynamical system method for solving monotone inverse variational inequalities (IVIs) in infinite dimensional Hilbert spaces. It is shown that the corresponding Cauchy problem admits a unique strong global solution, whose limit at infinity exists and solves the given monotone IVI. Then by discretizing the dynamical system, we obtain a class of iterative regularization algorithms with relaxation parameters, which are strongly convergent under quite mild assumptions on the cost operator. Some simple numerical examples, including an infinite dimensional one, are given to illustrate the performance of the proposed algorithms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availibility
The data that support the findings of this study are available from the corresponding author, upon reasonable request.
References
Chen JW, Ju XX, Kobis E, Liou YC (2020) Tikhonov type regularization methods for inverse mixed variational inequalities. Optimization 69:403–413
Dey S, Reich S (2023) A dynamical system for solving inverse quasi-variational inequalities. Optimization. https://doi.org/10.1080/02331934.2023.2173525
Goebel K, Reich S (1984) Uniform convexity. Hyperbolic geometry and nonexpansive mappings. Marcel Dekker, New York
Han QM, He BS (1998) A predict-correct method for a variant monotone variational inequality problem. Chin Sci Bull 43:1264–1267
Haraux A (1991) Systémes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées, Masson
He BS, Liu HX (2006) Inverse variational inequalities in the economic field: applications and algorithms, Sciencepaper Online. http://www.paper.edu.cn/downloadpaper.php? number=200609-260 (in Chinese)
He BS (1999) Inexact implicit methods for monotone general variational inequalities. Math Program 86:199–217
He S, Dong Q-L (2018) An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities. J Inequal Appl 2018:351
He XZ, Liu HX (2011) Inverse variational inequalities with projection-based solution methods. Eur J Oper Res 208:12–18
He BS, He XZ, Liu HX (2010) Sloving a class of constrained “blak-box’’ inverse vaiational inequalities. Eur J Oper Res 204:391–401
Ju XX, Li CD, He X, Feng G (2021) A proximal neurodynamic model for solving inverse mixed variational inequalities. Neural Netw. 138:1–9
Konnov IV (2000) Combined relaxation methods for variational inequalities. Springer, Berlin
Li X, Zou YZ (2016) Existence result and error bounds for a new class of inverse mixed quasi variational inequalities. J Inequal Appl 2016:42
Luo XP, Yang J (2014) Regularization and iterative methods for monotone inverse variational inequalities. Optim Lett 8:1261–1272
Noor MA, Noor KI, Khan AG (2015) Dynamical systems for quasi variational inequalities. Ann Funct Anal 6:193–209
Scrimali L (2012) An inverse variational inequality approach to the evolutionary spatial price equilibrium problem. Optim Eng 13:375–387
Tangkhawiwetkul J (2023) A neural network for solving the generalized inverse mixed variational inequality problem in Hilbert spaces. AIMS Math. 8:7258–7276
Vuong PT, He X, Thong DV (2021) Global exponential stability of a neural network for inverse variational inequalities. J Optim Theory Appl 190:915–930
Xu H (2002) Another control condition in an iterative method for nonexpansive mappings. Bull Austra Math Soc 65:109–113
Xu H-K, Dey S, Vetrivel V (2021) Notes on a neural network approach to inverse variational inequalities. Optimization 70:901–910
Zou X, Gong D, Wang L, Chen Z (2016) A novel method to solve inverse variational inequality problems based on neural networks. Neurocomputing 173:1163–1168
Acknowledgements
The authors thank the anonymous peer reviewers and the editor for their constructive comments which helped to improve the paper.
Funding
The research of T.N. Hai is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2021.02.
Author information
Authors and Affiliations
Contributions
P.K. Anh and T.N. Hai contributed to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript.
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Ethical approval
Not applicable.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
Additional information
Dedicated to Professor Le Dung Muu on the occasion of his 75th birthday with admiration and respect.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Anh, P.K., Hai, T.N. Regularized dynamics for monotone inverse variational inequalities in hilbert spaces. Optim Eng (2024). https://doi.org/10.1007/s11081-024-09882-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11081-024-09882-8
Keywords
- Inverse variational inequality
- Bilevel variational inequality
- Regularized dynamics
- Monotonicity
- Lipschitz continuity