连续零化神经网络(CTZNN)求解动态等式约束的线性规划问题

野性酷女 2022-12-26 11:18 122阅读 0赞

带动态等式约束的线性规划问题定义：

![\\large \\text\{minimize\}~C^\{\\text\{T\}\}(t)x(t)+d(t)][large _text_minimize_C_text_T_t_x_t_d_t]

![\\large \\text\{subject to\} ~ A(t)x(t)=b(t)][large _text_subject to_ _ A_t_x_t_b_t]

拉格朗日乘子法，将带线性约束的规划问题转化为无约束线性等式问题：

![\\large L(x(t), \\rho(t), t)=C^\{\\text\{T\}\}(t)x(t)+d(t)+\\rho^\{\\text\{T\}\}(t)(A(t)x(t)-b(t))][large L_x_t_ _rho_t_ t_C_text_T_t_x_t_d_t_rho_text_T_t_A_t_x_t_-b_t]

求解动态等式约束的线性规划问题，可以通过对下方的方程进行零化来实现：

![\\large \\frac\{\\partial L(x(t), \\rho(t), t)\}\{\\partial x(t)\}=C^\{\\text\{T\}\}(t)+\\rho(t)x(t)=0][large _frac_partial L_x_t_ _rho_t_ t_partial x_t_C_text_T_t_rho_t_x_t_0]

![\\large \\frac\{\\partial L(x(t), \\rho(t), t)\}\{\\partial \\rho(t)\}=A(t)x(t)-b(t)=0][large _frac_partial L_x_t_ _rho_t_ t_partial _rho_t_A_t_x_t_-b_t_0]

通过构造如下矩阵和向量，可以将以上方程组转化为动态线性方程：

![\\large \\begin\{bmatrix\} C^\{\\text\{T\}\}(t) &\\rho^\\text\{T\}(t) \\\\ A(t) & 0 \\end\{bmatrix\} \\times \\begin\{bmatrix\} I \\\\ x(t) \\end\{bmatrix\} =\\begin\{bmatrix\} 0\\\\ b(t) \\end\{bmatrix\}][large _begin_bmatrix_ C_text_T_t_ _rho_text_T_t_ _ A_t_ _ 0 _end_bmatrix_ _times _begin_bmatrix_ I _ x_t_ _end_bmatrix_ _begin_bmatrix_ 0_ b_t_ _end_bmatrix]

即为：

![\\large W(t)y(t)=u(t)][large W_t_y_t_u_t]

构造误差函数：

![\\large \\epsilon(t)=W(t)x(t)-b(t)][large _epsilon_t_W_t_x_t_-b_t]

零化神经网络的演化定义：

![\\large \\dot \\epsilon(t)=-\\gamma \\epsilon(t)][large _dot _epsilon_t_-_gamma _epsilon_t]

展开并整理可以得到用于求解态等数约束的线性规划问题的CTZNN模型迭代公式：

![\\large W(t)\\dot y(t)=-\\dot W(t)y(t)+\\dot u(t)-\\gamma(W(t)y(t)-u(t))][large W_t_dot y_t_-_dot W_t_y_t_dot u_t_-_gamma_W_t_y_t_-u_t]

[large _text_minimize_C_text_T_t_x_t_d_t]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Ctext%7Bminimize%7D~C%5E%7B%5Ctext%7BT%7D%7D%28t%29x%28t%29&plus;d%28t%29
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[large _frac_partial L_x_t_ _rho_t_ t_partial _rho_t_A_t_x_t_-b_t_0]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Cfrac%7B%5Cpartial%20L%28x%28t%29%2C%20%5Crho%28t%29%2C%20t%29%7D%7B%5Cpartial%20%5Crho%28t%29%7D%3DA%28t%29x%28t%29-b%28t%29%3D0
[large _begin_bmatrix_ C_text_T_t_ _rho_text_T_t_ _ A_t_ _ 0 _end_bmatrix_ _times _begin_bmatrix_ I _ x_t_ _end_bmatrix_ _begin_bmatrix_ 0_ b_t_ _end_bmatrix]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bbmatrix%7D%20C%5E%7B%5Ctext%7BT%7D%7D%28t%29%20%26%5Crho%5E%5Ctext%7BT%7D%28t%29%20%5C%5C%20A%28t%29%20%26%200%20%5Cend%7Bbmatrix%7D%20%5Ctimes%20%5Cbegin%7Bbmatrix%7D%20I%20%5C%5C%20x%28t%29%20%5Cend%7Bbmatrix%7D%20%3D%5Cbegin%7Bbmatrix%7D%200%5C%5C%20b%28t%29%20%5Cend%7Bbmatrix%7D
[large W_t_y_t_u_t]: https://private.codecogs.com/gif.latex?%5Clarge%20W%28t%29y%28t%29%3Du%28t%29
[large _epsilon_t_W_t_x_t_-b_t]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Cepsilon%28t%29%3DW%28t%29x%28t%29-b%28t%29
[large _dot _epsilon_t_-_gamma _epsilon_t]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Cdot%20%5Cepsilon%28t%29%3D-%5Cgamma%20%5Cepsilon%28t%29
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