| Citation: |
Jie Zhang, Wen Deng, Shuxin Wang. THE PROJECTION NEURAL NETWORK METHOD AND THE EULER METHOD FOR SOLVING THE SDLCP[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2392-2414. doi: 10.11948/20250277
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THE PROJECTION NEURAL NETWORK METHOD AND THE EULER METHOD FOR SOLVING THE SDLCP
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Abstract
This work introduces an innovative neural network based on projection techniques to address the Semi-Definite Linear Complementarity Problem (SDLCP). The SDLCP arises frequently in fields such as optimization theory, economic modeling, engineering computations, and operational analysis. From a theoretical standpoint, it is proven that, under appropriate assumptions, the developed method guarantees both the existence and uniqueness of solutions, alongside ensuring asymptotic and exponential stability. The continuous-time model is discretized using the explicit Euler scheme, and its convergence properties are rigorously established. To validate the proposed projection-based neural network approach and the discretization scheme, various MATLAB numerical tests are conducted. The designed neural network presents a viable and effective strategy for tackling SDLCP, showcasing considerable promise for practical deployment across disciplines.
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References
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Jie Zhang, Wen Deng, Shuxin Wang. THE PROJECTION NEURAL NETWORK METHOD AND THE EULER METHOD FOR SOLVING THE SDLCP[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2392-2414. doi: 10.11948/20250277
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Figure 1.
The evolution of
$ x_i(t) $ $ \alpha=0.02 $ -
Figure 2.
The evolution of
$ x_i(t) $ $ \alpha=0.0005 $ -
Figure 3.
The evolution of
$ x_i(t) $ $ \alpha=0.1 $ -
Figure 4.
Frobenius error
$ ||X_k - X_{k+1}|| $ -
Figure 5.
Heatmap visualization of the system.
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Figure 6.
Frobenius error ||Xk − Xk+1|| over iterations with varying δ and α.
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Figure 7.
Trajectories of x(t) over time.
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Figure 8.
||Xk − Xk+1|| over iterations.
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Figure 9.
Trajectories of x(t) over time.
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Figure 10.
||Xk − Xk+1|| over iterations.
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Figure 11.
Frobenius error ||Xk − Xk+1|| over iterations with varying δ and α.
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Figure 12.
Trajectories of the components of X(t).