带等式和不等式约束的线性分式规划Linear-fractional programming

以你之姓@ 2022-12-26 11:20 77阅读 0赞

## 1. 同时带等式和不等式约束的线性分式规划问题 ##

带等式和不等式约束的线性分式规划问题定义：

![\\large \\text\{minimize\} ~ f(x(t), t)=\\frac\{c^\{\\text\{T\}\}(t)x(t)+\\zeta(t)\}\{e^\{\\text\{T\}\}(t)x(t)+\\rho(t)\}][large _text_minimize_ _ f_x_t_ t_frac_c_text_T_t_x_t_zeta_t_e_text_T_t_x_t_rho_t]

![\\large \\large \\text\{dom\}~ f(x(t), t)=\\\{ x|e^\{\\text\{T\}\}(t)x(t)+\\rho(t)>0 \\\}][large _large _text_dom_ f_x_t_ t_ x_e_text_T_t_x_t_rho_t_0]

![\\large $$ \\text\{subject to\}\\left\\\{ \\begin\{aligned\} G(t)x(t) \\leqslant h(t) \\\\ A(t)x(t)=b(t) \\end\{aligned\} \\right. $$][large _ _text_subject to_left_ _begin_aligned_ G_t_x_t_ _leqslant h_t_ _ A_t_x_t_b_t_ _end_aligned_ _right.]

可以将以上的带等式和不等式约束的线性分式规划问题，转化为线性规划问题解决，

引入新变量，将目标函数f(x(t),t)转化为线性函数：

![\\large y(x(t),t)=\\frac\{x(t)\}\{e^\\text\{T\}(t)x(t)+\\rho(t)\}][large y_x_t_t_frac_x_t_e_text_T_t_x_t_rho_t]，![\\large z(x(t), t)=\\frac\{1\}\{e^\{\\text\{T\}\}(t)x(t)+\\rho(t)\}][large z_x_t_ t_frac_1_e_text_T_t_x_t_rho_t]。其中![\\large x(t)=\\frac\{y(x(t),t)\}\{z(x(t),t)\}][large x_t_frac_y_x_t_t_z_x_t_t]

将以上变换带入到原优化问题中可以得到如下等价的线性优化问题：

![\\large \\text\{minimize\}~ c^\{\\text\{T\}\}(t)y(x(t),t)+\\zeta(t)z(x(t),t)][large _text_minimize_ c_text_T_t_y_x_t_t_zeta_t_z_x_t_t]

![\\large \\text\{subject to\}~ $$ \\left\\\{ \\begin\{aligned\} G(t)y(x(t), t)-h(t)z(x(t),t)\\leqslant 0 \\\\ A(t)y(x(t),t)-b(t)z(x(t),t)=0 \\\\ e^\{\\text\{T\}\}(t)y(x(t),t)+\\rho(t)z(x(t),t)=1\\\\ z(x(t), t) \\geqslant 0 \\end\{aligned\} \\right. $$][large _text_subject to_ _ _left_ _begin_aligned_ G_t_y_x_t_ t_-h_t_z_x_t_t_leqslant 0 _ A_t_y_x_t_t_-b_t_z_x_t_t_0 _ e_text_T_t_y_x_t_t_rho_t_z_x_t_t_1_ z_x_t_ t_ _geqslant 0 _end_aligned_ _right.]

其中y，z为优化变量

## 2. 仅带等式约束的线性分式规划问题 ##

带等式约束的线性分式规划问题定义：

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

![\\large \\large \\text\{dom\}~ f(x(t), t)=\\\{ x|e^\{\\text\{T\}\}(t)x(t)+\\rho(t)>0 \\\}][large _large _text_dom_ f_x_t_ t_ x_e_text_T_t_x_t_rho_t_0]

引入新变量，将目标函数f(x(t),t)转化为线性函数：

将以上变换带入到原优化问题中可以得到如下等价的线性优化问题：

![\\large \\text\{minimize\}~ c^\{\\text\{T\}\}(t)y(x(t),t)+\\zeta(t)z(x(t),t)][large _text_minimize_ c_text_T_t_y_x_t_t_zeta_t_z_x_t_t]

![\\large \\text\{subject to\}~ $$ \\left\\\{ \\begin\{aligned\} A(t)y(x(t),t)-b(t)z(x(t),t)=0 \\\\ e^\{\\text\{T\}\}(t)y(x(t),t)+\\rho(t)z(x(t),t)=1 \\end\{aligned\} \\right. $$][large _text_subject to_ _ _left_ _begin_aligned_ A_t_y_x_t_t_-b_t_z_x_t_t_0 _ e_text_T_t_y_x_t_t_rho_t_z_x_t_t_1 _end_aligned_ _right.]

其中y，z为优化变量

[large _text_minimize_ _ f_x_t_ t_frac_c_text_T_t_x_t_zeta_t_e_text_T_t_x_t_rho_t]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Ctext%7Bminimize%7D%20~%20f%28x%28t%29%2C%20t%29%3D%5Cfrac%7Bc%5E%7B%5Ctext%7BT%7D%7D%28t%29x%28t%29&plus;%5Czeta%28t%29%7D%7Be%5E%7B%5Ctext%7BT%7D%7D%28t%29x%28t%29&plus;%5Crho%28t%29%7D
[large _large _text_dom_ f_x_t_ t_ x_e_text_T_t_x_t_rho_t_0]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Clarge%20%5Ctext%7Bdom%7D~%20f%28x%28t%29%2C%20t%29%3D%5C%7B%20x%7Ce%5E%7B%5Ctext%7BT%7D%7D%28t%29x%28t%29&plus;%5Crho%28t%29%3E0%20%5C%7D
[large _ _text_subject to_left_ _begin_aligned_ G_t_x_t_ _leqslant h_t_ _ A_t_x_t_b_t_ _end_aligned_ _right.]: https://private.codecogs.com/gif.latex?%5Clarge%20%24%24%20%5Ctext%7Bsubject%20to%7D%5Cleft%5C%7B%20%5Cbegin%7Baligned%7D%20G%28t%29x%28t%29%20%5Cleqslant%20h%28t%29%20%5C%5C%20A%28t%29x%28t%29%3Db%28t%29%20%5Cend%7Baligned%7D%20%5Cright.%20%24%24
[large y_x_t_t_frac_x_t_e_text_T_t_x_t_rho_t]: https://private.codecogs.com/gif.latex?%5Clarge%20y%28x%28t%29%2Ct%29%3D%5Cfrac%7Bx%28t%29%7D%7Be%5E%5Ctext%7BT%7D%28t%29x%28t%29&plus;%5Crho%28t%29%7D
[large z_x_t_ t_frac_1_e_text_T_t_x_t_rho_t]: https://private.codecogs.com/gif.latex?%5Clarge%20z%28x%28t%29%2C%20t%29%3D%5Cfrac%7B1%7D%7Be%5E%7B%5Ctext%7BT%7D%7D%28t%29x%28t%29&plus;%5Crho%28t%29%7D
[large x_t_frac_y_x_t_t_z_x_t_t]: https://private.codecogs.com/gif.latex?%5Clarge%20x%28t%29%3D%5Cfrac%7By%28x%28t%29%2Ct%29%7D%7Bz%28x%28t%29%2Ct%29%7D
[large _text_minimize_ c_text_T_t_y_x_t_t_zeta_t_z_x_t_t]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Ctext%7Bminimize%7D~%20c%5E%7B%5Ctext%7BT%7D%7D%28t%29y%28x%28t%29%2Ct%29&plus;%5Czeta%28t%29z%28x%28t%29%2Ct%29
[large _text_subject to_ _ _left_ _begin_aligned_ G_t_y_x_t_ t_-h_t_z_x_t_t_leqslant 0 _ A_t_y_x_t_t_-b_t_z_x_t_t_0 _ e_text_T_t_y_x_t_t_rho_t_z_x_t_t_1_ z_x_t_ t_ _geqslant 0 _end_aligned_ _right.]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Ctext%7Bsubject%20to%7D~%20%24%24%20%5Cleft%5C%7B%20%5Cbegin%7Baligned%7D%20G%28t%29y%28x%28t%29%2C%20t%29-h%28t%29z%28x%28t%29%2Ct%29%5Cleqslant%200%20%5C%5C%20A%28t%29y%28x%28t%29%2Ct%29-b%28t%29z%28x%28t%29%2Ct%29%3D0%20%5C%5C%20e%5E%7B%5Ctext%7BT%7D%7D%28t%29y%28x%28t%29%2Ct%29&plus;%5Crho%28t%29z%28x%28t%29%2Ct%29%3D1%5C%5C%20z%28x%28t%29%2C%20t%29%20%5Cgeqslant%200%20%5Cend%7Baligned%7D%20%5Cright.%20%24%24
[large _text_subject to_ _ A_t_x_t_b_t]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Ctext%7Bsubject%20to%7D%20~%20A%28t%29x%28t%29%3Db%28t%29
[large _text_subject to_ _ _left_ _begin_aligned_ A_t_y_x_t_t_-b_t_z_x_t_t_0 _ e_text_T_t_y_x_t_t_rho_t_z_x_t_t_1 _end_aligned_ _right.]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Ctext%7Bsubject%20to%7D~%20%24%24%20%5Cleft%5C%7B%20%5Cbegin%7Baligned%7D%20A%28t%29y%28x%28t%29%2Ct%29-b%28t%29z%28x%28t%29%2Ct%29%3D0%20%5C%5C%20e%5E%7B%5Ctext%7BT%7D%7D%28t%29y%28x%28t%29%2Ct%29&plus;%5Crho%28t%29z%28x%28t%29%2Ct%29%3D1%20%5Cend%7Baligned%7D%20%5Cright.%20%24%24