Stability of solution maps to mixed variational inequality
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Abstract
In this paper, the vector mixed variational inequality is considered, and the stability of the solution is studied in cases where both the objective function and the constraint set are perturbed. The Gerstewitz function is used to establish sufficient conditions for the Hausdorff continuity of the solution mapping of the above problem. An illustrative example is also provided to demonstrate the main results of the paper.
Tóm tắt
Trong bài báo này, bài toán bất đẳng thức biến phân vector hỗn hợp được xét và tính ổn định của nghiệm được nghiên cứu trong trường hợp cả hàm mục tiêu và tập ràng buộc đều bị nhiễu. Hàm Gerstewitz được sử dụng để thiết lập các điều kiện đủ cho tính liên tục theo nghĩa Hausdorff của ánh xạ nghiệm bài toán trên. Một ví dụ áp dụng cũng được đưa ra để minh họa cho kết quả chính của bài báo.
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References
Anh, L.Q., Khanh, P.Q., & Tam, T.N. (2019). Continuity of approximate solution maps of primal and dual vector equilibrium problems. Optimization Letters, 13, 201-211.
Anh, L.Q., Duoc, P.T., Tam, T.N., & Thang, N.C. (2021). Stability analysis for set-valued equilibrium problems with applications to Browder variational inclusions. Optimization Letters, 15, 613-626.
Anh, L.Q., Duoc, P.T., & Tung, N.M. (2022). On Lipschitz continuity of solutions to equilibrium problems via the Hiriart-Urruty oriented distance function. Computational and Applied Mathematics, 41(1),1-17.
Anh, L.Q., Tam, T.N., & Danh, N.H. (2023). On Lipschitz continuity of approximate solutions to set-valued equilibrium problems via nonlinear scalarization. Optimization, 72(2), 439–461.
Aubin J., & Frankowska H. (1990). Set-valued analysis. Springer Science & Business Media.
Boonman, P., Anh, L.Q., & Wangkeeree, R. (2021). Levitin-Polyak well-posedness by perturbations of strong vector mixed quasivariational inequality problems. Journal of Nonlinear and Convex Analysis, 22,1327-1352.
Cheng, Y., & Zhu, D. (2005). Global stability results for the weak vector variational inequality. Journal of Global Optimization, 32(4), 543–550.
Goeleven, D. (2017). Complementarity and variational inequalities in electronics. Academic Press.
Göpfert, A., Riahi, H., Tammer, C., & Zălinescu, C. (2003). Variational methods in partially ordered spaces. Springer Verlag, New York.
Iusem, A., & Lara, F. (2019). Existence results for noncoercive mixed variational inequalities in finite dimensional spaces. Journal of Optimization Theory and Applications, 183(1), 122–138.
Khan, A., Tammer, C., & Zălinescu, C. (2015). Set-valued optimization: An introduction with applications, Springer, Berlin.
Kinderlehrer, D., & Stampacchia, G. (2000). An introduction to variational inequalities and their applications. Society for Industrial and Applied Mathematics.
Li, S., & Chen, C. (2009). Stability of weak vector variational inequality. Nonlinear Analysis: Theory, Methods & Applications, 70(4), 1528–1535.
Mordukhovich, B.S. (2018). Variational analysis and applications. Springer.
Tanaka, T. (1997). Generalized semicontinuity and existence theorems for cone saddle points. Applied Mathematics and Optimization, 36(3), 313–322.
Tang, G.J., & Huang, N.J. (2013). Gap functions and global error bounds for set-valued mixed variational inequalities. Taiwanese Journal of Mathematics, 17(4), 1267-1286. DOI: 10.11650/tjm.17.2013.2247
Tang, G.J., & Li, Y.S. (2020). Existence of solutions for mixed variational inequalities with perturbation in Banach spaces. Optimization Letters, 14(3), 637–651.
Wang, M. (2017). The existence results and Tykhonov regularization method for generalized mixed variational inequalities in Banach spaces. Analysis and Mathematical Physics, 7(2), 151–163.
Yen, N.D. (1995). Hölder continuity of solutions to a parametric variational inequality. Applied Mathematics and Optimization, 31(3), 245–255.
Yen, N.D, & Lee, G.M. (1997). Solution sensitivity of a class of variational inequalities. Journal of Mathematical Analysis and Applications, 215(1), 48–55.