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 11:20 122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

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