连续零化神经网络模型求解时变、时不变二次极小化问题(Time-Variant Quadratic Minimization)

客官°小女子只卖身不卖艺 2022-11-15 11:22 149阅读 0赞

# 1. ZNN求解时变二次极小化问题： #

时变二次极小化问题的定义：

![\\text\{min\} ~~ \\frac\{1\}\{2\}x^T(t)Q(t)x(t)+p^T(t)x(t)][text_min_ _ _frac_1_2_x_T_t_Q_t_x_t_p_T_t_x_t]

![\\text\{Let\}~~M(t,x)=\\frac\{1\}\{2\}x^T(t)Q(t)x(t)+p^T(t)x(t)][text_Let_M_t_x_frac_1_2_x_T_t_Q_t_x_t_p_T_t_x_t]

M(t,x)的梯度可以写为如下形式：

![\\nabla M(t,x)=\\frac\{\\partial M(t,x)\}\{\\partial x\}=Q(t)x(t)+p(t).][nabla M_t_x_frac_partial M_t_x_partial x_Q_t_x_t_p_t_.]

显然当![\\nabla M(t,x)=0][nabla M_t_x_0]时，即可得到问题的理论解，因此需要构造：

![Q(t)x(t)+p(t)=0][Q_t_x_t_p_t_0]

进一步构造：

![\\epsilon (t)=Q(t)x(t)+p(t).][epsilon _t_Q_t_x_t_p_t_.]

根据积分增强ZNN的定义公式：

![\\dot\\epsilon (t)=-\\gamma \\epsilon(t)-\\mu\\int\_0^t\\epsilon(\\tau)d\\tau.][dot_epsilon _t_-_gamma _epsilon_t_-_mu_int_0_t_epsilon_tau_d_tau.]

可以得到最终用于求解时变二次极小化问题的ZNN模型：

![Q(t)\\dot x(t)=-\\dot Q(t)x(t)-\\dot p(t)-\\gamma (Q(t)x(t)+p(t))-\\mu\\int\_0^t (Q(\\tau)x(\\tau)+p(\\tau))d\\tau][Q_t_dot x_t_-_dot Q_t_x_t_-_dot p_t_-_gamma _Q_t_x_t_p_t_-_mu_int_0_t _Q_tau_x_tau_p_tau_d_tau]

# 2. ZNN求解时不变二次极小化问题： #

时不变二次极小化问题的定义：

![\\text\{min\} ~~ \\frac\{1\}\{2\}x^T(t)Qx(t)+p^Tx(t)][text_min_ _ _frac_1_2_x_T_t_Qx_t_p_Tx_t]

![\\text\{Let\}~~M(t,x)=\\frac\{1\}\{2\}x^T(t)Qx(t)+p^Tx(t)][text_Let_M_t_x_frac_1_2_x_T_t_Qx_t_p_Tx_t]

M(t,x)的梯度可以写为如下形式：

![\\nabla M(t,x)=\\frac\{\\partial M(t,x)\}\{\\partial x\}=Qx(t)+p.][nabla M_t_x_frac_partial M_t_x_partial x_Qx_t_p.]

显然当![\\nabla M(t,x)=0][nabla M_t_x_0]时，即可得到问题的理论解，因此需要构造：

![Qx(t)+p=0][Qx_t_p_0]

根据积分增强ZNN的定义公式，可以得到最终用于求解时变二次极小化问题的ZNN模型：

![Q\\dot x(t)=-\\gamma (Qx(t)+p)-\\mu\\int\_0^t (Qx(\\tau)+p)d\\tau][Q_dot x_t_-_gamma _Qx_t_p_-_mu_int_0_t _Qx_tau_p_d_tau]

[text_min_ _ _frac_1_2_x_T_t_Q_t_x_t_p_T_t_x_t]: https://latex.codecogs.com/gif.latex?%5Ctext%7Bmin%7D%20~~%20%5Cfrac%7B1%7D%7B2%7Dx%5ET%28t%29Q%28t%29x%28t%29&plus;p%5ET%28t%29x%28t%29
[text_Let_M_t_x_frac_1_2_x_T_t_Q_t_x_t_p_T_t_x_t]: https://latex.codecogs.com/gif.latex?%5Ctext%7BLet%7D~~M%28t%2Cx%29%3D%5Cfrac%7B1%7D%7B2%7Dx%5ET%28t%29Q%28t%29x%28t%29&plus;p%5ET%28t%29x%28t%29
[nabla M_t_x_frac_partial M_t_x_partial x_Q_t_x_t_p_t_.]: https://latex.codecogs.com/gif.latex?%5Cnabla%20M%28t%2Cx%29%3D%5Cfrac%7B%5Cpartial%20M%28t%2Cx%29%7D%7B%5Cpartial%20x%7D%3DQ%28t%29x%28t%29&plus;p%28t%29.
[nabla M_t_x_0]: https://latex.codecogs.com/gif.latex?%5Cnabla%20M%28t%2Cx%29%3D0
[Q_t_x_t_p_t_0]: https://latex.codecogs.com/gif.latex?Q%28t%29x%28t%29&plus;p%28t%29%3D0
[epsilon _t_Q_t_x_t_p_t_.]: https://latex.codecogs.com/gif.latex?%5Cepsilon%20%28t%29%3DQ%28t%29x%28t%29&plus;p%28t%29.
[dot_epsilon _t_-_gamma _epsilon_t_-_mu_int_0_t_epsilon_tau_d_tau.]: https://latex.codecogs.com/gif.latex?%5Cdot%5Cepsilon%20%28t%29%3D-%5Cgamma%20%5Cepsilon%28t%29-%5Cmu%5Cint_0%5Et%5Cepsilon%28%5Ctau%29d%5Ctau.
[Q_t_dot x_t_-_dot Q_t_x_t_-_dot p_t_-_gamma _Q_t_x_t_p_t_-_mu_int_0_t _Q_tau_x_tau_p_tau_d_tau]: https://latex.codecogs.com/gif.latex?Q%28t%29%5Cdot%20x%28t%29%3D-%5Cdot%20Q%28t%29x%28t%29-%5Cdot%20p%28t%29-%5Cgamma%20%28Q%28t%29x%28t%29&plus;p%28t%29%29-%5Cmu%5Cint_0%5Et%20%28Q%28%5Ctau%29x%28%5Ctau%29&plus;p%28%5Ctau%29%29d%5Ctau
[text_min_ _ _frac_1_2_x_T_t_Qx_t_p_Tx_t]: https://latex.codecogs.com/gif.latex?%5Ctext%7Bmin%7D%20~~%20%5Cfrac%7B1%7D%7B2%7Dx%5ET%28t%29Qx%28t%29&plus;p%5ETx%28t%29
[text_Let_M_t_x_frac_1_2_x_T_t_Qx_t_p_Tx_t]: https://latex.codecogs.com/gif.latex?%5Ctext%7BLet%7D~~M%28t%2Cx%29%3D%5Cfrac%7B1%7D%7B2%7Dx%5ET%28t%29Qx%28t%29&plus;p%5ETx%28t%29
[nabla M_t_x_frac_partial M_t_x_partial x_Qx_t_p.]: https://latex.codecogs.com/gif.latex?%5Cnabla%20M%28t%2Cx%29%3D%5Cfrac%7B%5Cpartial%20M%28t%2Cx%29%7D%7B%5Cpartial%20x%7D%3DQx%28t%29&plus;p.
[Qx_t_p_0]: https://latex.codecogs.com/gif.latex?Qx%28t%29&plus;p%3D0
[Q_dot x_t_-_gamma _Qx_t_p_-_mu_int_0_t _Qx_tau_p_d_tau]: https://latex.codecogs.com/gif.latex?Q%5Cdot%20x%28t%29%3D-%5Cgamma%20%28Qx%28t%29&plus;p%29-%5Cmu%5Cint_0%5Et%20%28Qx%28%5Ctau%29&plus;p%29d%5Ctau