# American Institute of Mathematical Sciences

January  2011, 7(1): 183-198. doi: 10.3934/jimo.2011.7.183

## A differential equation method for solving box constrained variational inequality problems

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 School of Sciences, Dalian Nationalities University, Dalian, 116066, China 3 Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, LiaoNing

Received  April 2010 Revised  October 2010 Published  January 2011

In this paper, we discuss a system of differential equations based on the projection operator for solving the box constrained variational inequality problems. The equilibrium solutions to the differential equation system are proved to be the solutions of the box constrained variational inequality problems. Two differential inclusion problems associated with the system of differential equations are introduced. It is proved that the equilibrium solution to the differential equation system is locally asymptotically stable by verifying the locally asymptotical stability of the equilibrium positions of the differential inclusion problems. An Euler discrete scheme with Armijo line search rule is introduced and its global convergence is demonstrated. The numerical experiments are reported to show that the Euler method is effective.
Citation: Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
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##### References:
 [1] Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200 [2] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [3] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 [4] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [5] Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 [6] Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 [7] Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 [8] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [9] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [10] Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 [11] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [12] Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 [13] Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 [14] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [15] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [16] Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 [17] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [18] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [19] Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008 [20] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

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