A new approach to equilibrium problems via regularized extragradient-type dynamics
Email:
hanguyenhai@utc.edu.vn
Từ khóa:
equilibrium problem, dynamical system, iterative method, regularization
Tóm tắt
Nonlinear analysis serves as a fundamental framework for modeling diverse phenomena in modern science and engineering. Within this field, equilibrium problems provide a powerful tool that not only unifies diverse mathematical models, such as variational inequalities and fixed point problems, but also effectively characterizes various phenomena in various scientific disciplines. Over the past decades, dynamical systems have emerged as an effective method for finding solutions to these problems due to their continuous-time nature and stability properties. This paper investigates a novel regularized dynamical system of extragradient-type specifically designed to solve monotone equilibrium problems. The primary challenge addressed is ensuring the existence of a global solution and achieving strong convergence under mild assumptions. Our methodological approach involves constructing a continuous-time trajectory governed by a regularization term and an extragradient-type structure. We first prove the existence and uniqueness of a strong global solution to the proposed system. Subsequently, the asymptotic behavior of the system is analyzed, demonstrating the strong convergence of the generated trajectories toward a specific solution of the original equilibrium problem. Lastly, a time discretization scheme is implemented, leading to a new iterative regularized extragradient-type algorithm enhanced by relaxation parameters. The main results confirm that the proposed dynamical approach not only guarantees theoretical stability but also provides a robust computational platform for monotone equilibrium problems. The findings of this study offer significant insights into the development of efficient continuous-time optimization methods and contribute to the broader understanding of regularization techniques in nonlinear analysis.Tài liệu tham khảo
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[12]. P.K. Anh, T.N. Hai, V.T. Dung, A gradient-like regularized dynamics for monotone equilibrium problems, Qualitative Theory Dyn. Sys, 21(160) (2022). https://doi.org/10.1007/s12346-022-00698-4
[13]. N.T. Thuong, N.T. Vinh, Convergence of one-step projection methods for equilibrium problems given by a sum of two bifunctions, Vietnam J. Math., 54 (2026), 523-547.
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[15]. A. Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées, Masson, Paris, 1991.
[16]. N. H. Ha, Supplementary proofs of REA algorithms.
https://anonymous.4open.science/r/Proof-31E8/Proof.pdf (accessed June 2, 2026).
[2]. A.S. Antipin, Gradient approach of computing fixed points of equilibrium problems, J. Glob. Optim, 24 (2002) 285-309. https://doi.org/10.1023/A:1020321209606
[3]. I.V. Konnov, Equilibrium Models and Variational Inequalities, Elsevier, Amsterdam, 2007.
[4]. A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geometry, 15 (1999) 91-100.
[5]. T.D. Quoc, L.D. Muu, V.H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008) 749-776. https://doi.org/10.1080/02331930601122876
[6]. P.T. Vuong, J.J. Strodiot, V.H. Nguyen, On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space, Optimization, 64 (2015) 429-451. https://doi.org/10.1080/02331934.2012.759327
[7]. D.V. Hieu, L.D. Muu, P.K. Quy, Regularization iterative method of bilevel form for equilibrium problems in Hilbert spaces, Math. Meth. Appl. Sci, 45(10) (2022) 6143-6164. https://doi.org/10.1002/mma.8162
[8]. S. Banert, R.I. Bot, A forward–backward–forward differential equation and its asymptotic properties, J. Convex Anal, 25(2) (2018) 371-388. https://doi.org/10.48550/arXiv.1503.07728
[9]. T.N. Hai, Dynamical systems for solving variational inequalities, J. Dyn. Control Syst, 28 (2022) 681-696. https://doi.org/10.1007/s10883-021-09531-8
[10]. P.T. Vuong, J.J. Strodiot, A dynamical system for strongly pseudo-monotone equilibrium problems, J. Optim. Theory Appl, 185 (2020) 767-784. https://doi.org/10.1007/s10957-020-01669-y
[11]. L.V. Vinh, V.N. Tran, P.T. Vuong, A second-order dynamical system for equilibrium problems, Numer. Algor, 91 (2022) 327-351. https://doi.org/10.1007/s11075-022-01264-4
[12]. P.K. Anh, T.N. Hai, V.T. Dung, A gradient-like regularized dynamics for monotone equilibrium problems, Qualitative Theory Dyn. Sys, 21(160) (2022). https://doi.org/10.1007/s12346-022-00698-4
[13]. N.T. Thuong, N.T. Vinh, Convergence of one-step projection methods for equilibrium problems given by a sum of two bifunctions, Vietnam J. Math., 54 (2026), 523-547.
[14]. H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc, 66 (2002) 240-256. https://doi.org/10.1112/S0024610702003332
[15]. A. Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées, Masson, Paris, 1991.
[16]. N. H. Ha, Supplementary proofs of REA algorithms.
https://anonymous.4open.science/r/Proof-31E8/Proof.pdf (accessed June 2, 2026).
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02/06/2026
Nhận bài sửa
28/10/2026
Chấp nhận đăng
04/11/2026
Xuất bản
15/11/2026
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Kiểu trích dẫn
Nguyen Hai, H. (1794675600). A new approach to equilibrium problems via regularized extragradient-type dynamics. Tạp Chí Khoa Học Giao Thông Vận Tải, 77(9), 1155-1169. https://doi.org/10.47869/tcsj.77.9.3
