【凸优化】分式规划

P 1 : min ⁡ c , f , W , ν α β ≥ Γ , ∣ ν m c m ∣ 2 + W m , m ≤ P , m ∈ M , W ⪰ 0 , ν m ∈ { 0 , 1 } , m ∈ M , \begin{aligned} \mathcal{P}1:\min_{\boldsymbol{c},\boldsymbol{f},\boldsymbol{W},\boldsymbol{\nu}} &\alpha \\ & \beta\geq\Gamma, \\ & |\nu_{m}c_{m}|^{2}+W_{m,m}\leq P,m\in\mathcal{M}, \\ & W\succeq0, \\ & \nu_{m}\in\{0,1\},m\in\mathcal{M}, \end{aligned} P1:c,f,W,νmin​​αβ≥Γ,∣νm​cm​∣2+Wm,m​≤P,m∈M,W⪰0,νm​∈{0,1},m∈M,​

$\begin{array}{rl}\alpha & =\sum _{m=1}^M{\nu }_m^2-\frac{f^{†}\left({\sum }_{m=1}^M{c}_m{\nu }_m{h}_m\right){\left({\sum }_{m=1}^M{c}_m{\nu }_m{h}_m\right)}^{†}f}{f^{†}\left({\sum }_{m=1}^M|c_m{|}^2{\nu }_m^2{h}_m{h}_m^{†}+HW{H}^{†}+A\right)f}\end{array}$

$\begin{array}{rl}\beta & =\sum _{m=1}^M{\nu }_m^2-\frac{|{\sum }_{m=1}^M{c}_m{\nu }_m{g}_m{|}^2}{{\sum }_{m=1}^M|c_m{|}^2{\nu }_m^2|g_m{|}^2+g^{†}Wg+P_{\mathrm{J}}|g_{\mathrm{J}}{|}^2+{\sigma }_{\mathrm{e}}^2}\end{array}$

定义辅助变量$\tilde{\mathbf{\nu }}$来处理离散的整数变量$\mathbf{\nu }$

P 2 : min ⁡ f , c , W , ν ~ , ν α  s.t.  β ≥ Γ , ν ~ m = ν m , m ∈ M , ν m ( 1 − ν ~ m ) = 0 , m ∈ M , c m ( 1 − ν m ) = 0 , m ∈ M , W ⪰ 0 , ν m ∈ { 0 , 1 } , m ∈ M , \begin{aligned} \mathcal{P} 2: \min _{\boldsymbol{f}, c, \boldsymbol{W}, \tilde{\boldsymbol{\nu}}, \boldsymbol{\nu}} & \alpha \\ \text { s.t. } & \beta \geq \Gamma, \\ & \tilde{\nu}_m=\nu_m, \quad m \in \mathcal{M}, \\ & \nu_m\left(1-\tilde{\nu}_m\right)=0, \quad m \in \mathcal{M}, \\ & c_m\left(1-\nu_m\right)=0, \quad m \in \mathcal{M}, \\ & W\succeq0, \\ & \nu_{m}\in\{0,1\},m\in\mathcal{M},\end{aligned} P2:f,c,W,ν~,νmin​ s.t. ​αβ≥Γ,ν~m​=νm​,m∈M,νm​(1−ν~m​)=0,m∈M,cm​(1−νm​)=0,m∈M,W⪰0,νm​∈{0,1},m∈M,​

下面利用罚对偶分解（penalty dual decomposition）来处理上面的优化问题，首先定义增广拉格朗日函数

$\begin{array}{rl}& \mathcal{A}=\alpha +\psi =\alpha +{\mathbf{\kappa }}^{\top }(\mathbf{\nu }-\tilde{\mathbf{\nu }})+\sum _{m=1}^M{\varpi }_m\left({\nu }_m\left(1-{\tilde{\nu }}_m\right)\right)+\sum _{m=1}^M{\eta }_m\left(c_m\left(1-{\nu }_m\right)\right)+\frac{1}{2\varrho }\left(\Vert \mathbf{\nu }-\tilde{\mathbf{\nu }}{\Vert }^2\right)\\ & +\frac{1}{2\varrho }\left(\sum _{m=1}^M{\left({\nu }_m\left(1-{\tilde{\nu }}_m\right)\right)}^2+{\left|c_m\left(1-{\nu }_m\right)\right|}^2\right)\end{array}$
其中$\mathbf{\kappa }$ $\varpi$ $\nu$ $\eta$是对偶变量，$\varrho$是罚系数，于是求解下面的问题

$\begin{array}{rl}\mathcal{P}3:\underset{\begin{array}{c}\mathbf{f},\mathbf{c},\mathbf{W},\tilde{\mathbf{\nu }},\mathbf{\nu }\end{array},\mathbf{\kappa },\varpi ,\varrho ,\eta }{\min }& \mathcal{A},\\ \text{ s.t. }& |{\nu }_m{c}_m{|}^2+W_{m,m}\le P,m\in \mathcal{M},\\ & W⪰0,\\ & \beta \ge \Gamma \end{array}$

在罚对偶分解中外层循环更新$\mathbf{f},c,\mathbf{W},\tilde{\mathbf{\nu }},\mathbf{\kappa }$ $\varpi$ $\nu$ $\varrho$，内层更新下面的变量

当给定$c,\mathbf{W},\tilde{\mathbf{\nu }},\mathbf{\nu }$，最优的$\mathbf{f}$等价求解如下问题
$\mathcal{P}3.1:{\max }_{\mathbf{f}}\frac{{\mathbf{f}}^{†}\left({\sum }_{m=1}^M{c}_m{\mathbf{h}}_m\right){\left({\sum }_{m=1}^M{c}_m{\mathbf{h}}_m\right)}^{†}\mathbf{f}}{{\mathbf{f}}^{†}\left({\sum }_{m=1}^M{|c_m|}^2{\mathbf{h}}_m{\mathbf{h}}_m^{†}+\mathbf{F}\right)\mathbf{f}},$

基于广义瑞利熵 $\mathbf{F}=\mathbf{H}\mathbf{W}{\mathbf{H}}^{†}+\mathbf{A}$. 最优的$\mathbf{f}$ 如下
$\mathbf{f}={\left({\sum }_{m=1}^M{\left|c_m\right|}^2{\mathbf{h}}_m{\mathbf{h}}_m^{†}+\mathbf{F}\right)}^{-1}{\sum }_{m=1}^M{c}_m{\mathbf{h}}_m$

当给定 { f , W , ν ~ , ν } \{\boldsymbol{f}, \boldsymbol{W}, \tilde{\boldsymbol{\nu}}, \boldsymbol{\nu}\} {f,W,ν~,ν} 更新$\mathbf{c}$, 则问题退化为一个分式规划，使用二次转换（quadratic transform）忽略与$\mathbf{c}$无关的项，则$\mathcal{P}3$可以转为如下优化问题，其中$t$是辅助变量。

$\begin{array}{rl}\mathcal{P}3.2:{\max }_{\mathbf{c},t}& 2\Re \left\{t^{\ast }{\mathbf{f}}^{†}\mathbf{H}\mathbf{c}\right\}-|t{|}^2\left({\mathbf{c}}^{†}\mathbf{D}\mathbf{c}+\zeta \right)-\varphi \\ \text{ s.t. }& {\mathbf{c}}^{†}\mathbf{g}{\mathbf{g}}^{†}\mathbf{c}-\left(1^{\top }\mathbf{\nu }-\Gamma \right)\left({\mathbf{c}}^{†}\mathbf{B}\mathbf{c}+\varsigma \right)\le 0\\ & |{\nu }_m{c}_m{|}^2+W_{m,m}\le P,m\in \mathcal{M},\end{array}$

$\begin{array}{rl}& \varphi \triangleq (\mathbf{\eta }\odot (1-\mathbf{\nu }){)}^{\top }\mathbf{c}+\frac{1}{2\varrho }{\mathbf{c}}^{†}\mathrm{diag}((1-\mathbf{\nu })\odot (1-\mathbf{\nu }))\mathbf{c}\\ & \zeta \triangleq {\mathbf{f}}^{†}\left(\mathbf{H}\mathbf{W}{\mathbf{H}}^{†}+\mathbf{A}\right)\mathbf{f}\\ & \varsigma \triangleq {\mathbf{g}}^{†}\mathbf{W}\mathbf{g}+P_{\mathrm{J}}{\left|g_{\mathrm{J}}\right|}^2+{\sigma }_{\mathrm{e}}^2\\ & \mathbf{B}\triangleq \mathrm{diag}\left({\left[{\left|g_1\right|}^2,\dots ,{\left|g_M\right|}^2\right]}^{\top }\right)\\ & \mathbf{D}\triangleq \triangleq \mathrm{diag}\left({\left[f^{†}{\mathbf{h}}_1{\mathbf{h}}_1^{†}\mathbf{f},\dots ,{\mathbf{f}}^{†}{\mathbf{h}}_M{\mathbf{h}}_M^{†}\mathbf{f}\right]}^{\top }\right),\end{array}$