| Citation: |
Danna Jia, Li Wang, Juhe Sun, Huiting Zhuang. THE SECOND-ORDER DIFFERENTIAL EQUATION METHOD FOR SOLVING THE VARIATIONAL INEQUALITY PROBLEM[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 760-777. doi: 10.11948/20230084
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THE SECOND-ORDER DIFFERENTIAL EQUATION METHOD FOR SOLVING THE VARIATIONAL INEQUALITY PROBLEM
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Abstract
In this paper, we smooth the Karush-Kuhn-Tucker (KKT) functions for the classical variational inequality problem, then it is transformed to an unconstrained optimization problem. We firstly establish the second-order differential equation system involving two time-dependent parameters for solving the unconstrained optimization problem. The global convergence theorem for the second-order differential equation system is proved. At last, four numerical experiments are reported to verify the effectiveness of the second-order differential equation method for solving the classical variational inequality problem.
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References
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Danna Jia, Li Wang, Juhe Sun, Huiting Zhuang. THE SECOND-ORDER DIFFERENTIAL EQUATION METHOD FOR SOLVING THE VARIATIONAL INEQUALITY PROBLEM[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 760-777. doi: 10.11948/20230084
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Figure 1.
Trajectories of $ x\left(t \right) $ of the second-order differential equation system (3.1) for Example 5.1 from six random initial points about x.
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Figure 2.
Comparison of error rates of $ {\left\| {x\left(t \right) - {x^*}} \right\|_2} $ for the first-order system (4.1) and the second-order system (3.1) for Example 5.1.
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Figure 3.
Trajectories of $ x\left(t \right) $ of the second-order differential equation system (3.1) for Example 5.2 from five random initial points about x.
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Figure 4.
Comparison of error rates of $ {\left\| {x\left(t \right) - {x^*}} \right\|_2} $ for the first-order system (4.1) and the second-order system (3.1) for Example 5.2.
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Figure 5.
Trajectories of $ x\left(t \right) $ of the second-order differential equation system (3.1) for Example 5.3 from five random initial points about x.
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Figure 6.
Comparison of error rates of $ {\left\| {x\left(t \right) - {x^*}} \right\|_2} $ for the first-order system (4.1) and the second-order system (3.1) for Example 5.3.
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Figure 7.
Trajectories of $ x\left(t \right) $ of the second-order differential equation system (3.1) for Example 5.4 from three random initial points about x.
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Figure 8.
Comparison of error rates of $ {\left\| {x\left(t \right) - {x^*}} \right\|_2} $ for the first-order system (4.1) and the second-order system (3.1) for Example 5.4.