一类变分不等式问题的二阶微分方程方法
The Second Order Differential Equation Method for a Class of Variational Inequality Problem
摘要: 本文运用光滑化的自然残差函数建立了具有不等式约束条件的变分不等式问题的光滑化KKT方程组，并建立了与其等价的无约束优化问题。建立了具有阻尼惯性参数和时间尺度参数的二阶微分方程系统来求解该无约束优化问题，并证明了该二阶微分方程系统的稳定性，从而得到了具有不等式约束的变分不等式问题的KKT点的收敛性。并将二阶微分方程方法与已有的一阶微分方程方法进行了理论条件和数值结果的对比。在理论条件的要求上，二阶微分方程方法的条件要更容易实现，而在数值结果上，一阶微分方程方法的收敛速度要快，但是两种方法的差距可以忽略不计。
Abstract: In this paper, a smoothing KKT equation system with inequality constraints is established by using the smoothing natural residual function, and the unconstrained optimization problem is established. A second order differential equation system with damping coefficient and time scale coefficient is established to solve the unconstrained optimization problem, and the stability of the second-order differential equation system is proved, then the convergence of the KKT points of the variational inequality problem with inequality constraints is obtained. The theoretical conditions and numerical results of the second-order differential equation method are compared with the existing first order differential equation method. In terms of theoretical conditions, the conditions of the second-order differential equation method are easier to implement, while in the numerical results, the convergence speed of the first order differential equation method is faster, but the difference between the two methods is negligible.
文章引用：于晓雪, 安思源, 宋瑷如, 施玉缘, 唐博文, 陈宇航, 王莉. 一类变分不等式问题的二阶微分方程方法[J]. 应用数学进展, 2024, 13(12): 5439-5449. https://doi.org/10.12677/aam.2024.1312525

参考文献

[1] Facchinei, F. and Pang, J.S. (2003) Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer.
[2] Gao, X.-B., Liao, L.-Z. and Qi, L. (2005) A Novel Neural Network for Variational Inequalities with Linear and Nonlinear Constraints. IEEE Transactions on Neural Networks, 16, 1305-1317. [Google Scholar] [CrossRef] [PubMed]
[3] He, B.S. and Yang, H. (2000) A Neural Network Model for Monotone Linear Asymmetric Variational Inequalities. IEEE Transactions on Neural Networks, 11, 3-16. [Google Scholar] [CrossRef] [PubMed]
[4] Zhou, L.M., Zhang, L.W. and He, S.X. (2005) A Differential Equation Approach to Solving Nonlinear Complementarity Problems. OR Transactions, 9, 8-16.
[5] Liao, L., Qi, H. and Qi, L. (2001) Solving Nonlinear Complementarity Problems with Neural Networks: A Reformulation Method Approach. Journal of Computational and Applied Mathematics, 131, 343-359. [Google Scholar] [CrossRef] 
[6] Antipin, A.S. (2000) Solving Variational Inequalities with Coupling Constraints with the Use of Differential Equations. Differential Equations, 36, 1587-1596. [Google Scholar] [CrossRef] 
[7] 王莉. 变分不等式的微分方程方法与增广Lagrange方法[D]: [硕士学位论文]. 大连: 大连理工大学, 2011.
[8] Wang, L., Chen, X. and Sun, J. (2021) The Second-Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints. Complexity, 2021, 1-12. [Google Scholar] [CrossRef] 
[9] 王莉, 陈星旭, 孙菊贺, 杨峥. 二阶微分方程系统求解极值映射不动点问题[J]. 沈阳航空航天大学学报, 2022, 39(1): 77-84.
[10] Attouch, H., Chbani, Z. and Riahi, H. (2019) Fast Convex Optimization via Time Scaling of Damped Inertial Gradient Dynamics.
https://hal.archivesouvertes.fr/hal-02138954
[11] Attouch, H. and Cabot, A. (2017) Asymptotic Stabilization of Inertial Gradient Dynamics with Time-Dependent Viscosity. Journal of Differential Equations, 263, 5412-5458. [Google Scholar] [CrossRef] 
[12] Attouch, H., Chbani, Z., Peypouquet, J. and Redont, P. (2016) Fast Convergence of Inertial Dynamics and Algorithms with Asymptotic Vanishing Viscosity. Mathematical Programming, 168, 123-175. [Google Scholar] [CrossRef] 
[13] Attouch, H. and Peypouquet, J. (2019) Convergence Rate of Proximal Inertial Algorithms Associated with Moreau Envelopes of Convex Functions. In: Splitting Algorithms, Modern Operator Theory, and Applications, Springer, 1-44. [Google Scholar] [CrossRef] 
[14] Mosco, U. (1976) Implicit Variational Problems and Quasi Variational Inequalities. In: Lecture Notes in Mathematics, Springer, 83-156. [Google Scholar] [CrossRef] 
[15] Sun, J., Fu, W., Alcantara, J.H. and Chen, J. (2021) A Neural Network Based on the Metric Projector for Solving SOCCVI Problem. IEEE Transactions on Neural Networks and Learning Systems, 32, 2886-2900. [Google Scholar] [CrossRef] [PubMed]

为你推荐