Block Coordinate Descent算法的部分构造技巧

构造的目的

通过增加辅助变量，使原来的非凸问题变为关于各个变量的凸子问题，交替优化各个辅助变量。

定理

Define an $m$ by $m$ matrix function
$$\mathbf{E}(\mathbf{U},\mathbf{V})\triangleq \left(\mathbf{I}-{\mathbf{U}}^H\mathbf{HV}\right){\left(\mathbf{I}-{\mathbf{U}}^H\mathbf{HV}\right)}^H+{\mathbf{U}}^H\mathbf{NU}$$
where $\mathbf{N}$ is any positive definite matrix. The following three facts hold true.

1. $\text{ For any positive definite matrix }\mathbf{E}\in {\mathbb{C}}^{m\times m}\text{, we have }$
   $${\mathbf{E}}^{-1}=\arg \underset{\mathbf{W}\succ 0}{\max }\log \det (\mathbf{W})-\mathrm{Tr}(\mathbf{WE})$$
   （注：argmax表示找到使某个函数取得最大值的参数值）
   and
   $$-\log \det (\mathbf{E})=\underset{\mathbf{W}\succ 0}{\max }\log \det (\mathbf{W})-\mathrm{Tr}(\mathbf{WE})+m$$

2. $\text{ For any positive definite matrix }\mathbf{W}\text{, we have }$
   $$\begin{array}{rl}\tilde{\mathbf{U}}& \triangleq \arg \underset{\mathbf{U}}{\min }\mathrm{Tr}(\mathbf{WE}(\mathbf{U},\mathbf{V}))\\ & ={\left(\mathbf{N}+\mathbf{HVV}{\mathbf{H}}^H{\mathbf{H}}^H\right)}^{-1}\mathbf{HV}\end{array}$$
   and
   $$\begin{array}{rl}\mathbf{E}(\tilde{\mathbf{U}},\mathbf{V})& =\mathbf{I}-{\tilde{\mathbf{U}}}^H\mathbf{HV}\\ & ={\left(\mathbf{I}+{\mathbf{V}}^H{\mathbf{H}}^H{\mathbf{N}}^{-1}\mathbf{HV}\right)}^{-1}.\end{array}$$

3） We have
$$\begin{array}{rl}\log \det (\mathbf{I}+& \mathbf{HV}{\mathbf{V}}^H{\mathbf{H}}^H{\mathbf{N}}^{-1})\\ & =\underset{\mathbf{W}\succ 0,\mathbf{U}}{\max }\log \det (\mathbf{W})-\mathrm{Tr}(\mathbf{WE}(\mathbf{U},\mathbf{V}))+m\end{array}$$

Facts 1) and 2) can be proven by simply using the first-order optimality condition, while Fact 3) directly follows from Facts 1) and 2) and the identity $\log \det (\mathbf{I}+\mathbf{AB})=\log \det (\mathbf{I}+\mathbf{BA})$ . We refer readers to [32], [33] for more detailed proof.

Next, using Lemma 4.1, we derive an equivalent problem of problem (5) by introducing some auxiliary variables. Define
$$\mathbb{E}(\mathbf{U},\mathbf{V})\triangleq \left(\mathbf{I}-{\mathbf{U}}^H{\mathbf{H}}_I\mathbf{V}\right){\left(\mathbf{I}-{\mathbf{U}}^H{\mathbf{H}}_I\mathbf{V}\right)}^H+{\mathbf{U}}^H\mathbf{U}.$$

Then we have from Fact 3) that

$$\begin{array}{rl}\log \det (\mathbf{I}& +{\mathbf{H}}_I\mathbf{V}{\mathbf{V}}^H{\mathbf{H}}_I^H)\\ & =\underset{{\mathbf{W}}_I\succ 0,\mathbf{U}}{\max }\log \det \left({\mathbf{W}}_I\right)-\mathrm{Tr}\left({\mathbf{W}}_I\mathbb{E}(\mathbf{U},\mathbf{V})\right)+d\end{array}$$

Furthermore, from Fact 1), we have

$$\begin{array}{l}-\log \det \left(\mathbf{I}+{\mathbf{H}}_E\mathbf{V}{\mathbf{V}}^H{\mathbf{H}}_E^H\right)\\ ={\max }_{{\mathbf{W}}_E\succ 0}\log \det \left({\mathbf{W}}_E\right)-\mathrm{Tr}\left({\mathbf{W}}_E\left(\mathbf{I}+{\mathbf{H}}_E\mathbf{V}{\mathbf{V}}^H{\mathbf{H}}_E^H\right)\right)+N_E.\end{array}$$

另一篇中对于该定理的表述

Physical Layer Security in Near-Field Communications

该构造与WMMSE算法的关系

考虑一个全数字波束成形优化问题：
$$\begin{array}{ll}{\max }_{{\mathbf{W}}_{\mathrm{FD}}}& {\log }_2\left|{\mathbf{I}}_{M_{\mathrm{B}}}+{\sigma }_{\mathrm{B}}^{-2}{\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}{\mathbf{W}}_{\mathrm{FD}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}^{\mathrm{H}}\right|\\ \text{ s.t. }& {\Vert {\mathbf{W}}_{\mathrm{FD}}\Vert }_{\mathrm{F}}^2⩽P_{\max },\\ & {\Vert {\mathbf{H}}_{\mathrm{W}}{\mathbf{W}}_{\mathrm{FD}}\Vert }_{\mathrm{F}}^2⩽p_{\text{leak. }}.\end{array}\text{\hspace{1em}(26)}$$
基于 WMMSE 的算法被设计来解决这个子问题。其主要思想是通过利用速率最大化问题和均方误差（MSE）最小化问题之间的等价性，将原始问题转化为更容易处理的形式[38]。
Specifically, the signal vector $\tilde{\mathbf{s}}$ at Bob is estimated by an introduced linear receive beamforming matrix $\mathbf{U}\in {\mathbb{C}}^{M_{\mathrm{B}}\times L}$ as $\tilde{\mathbf{s}}={\mathbf{U}}^{\mathrm{H}}{\mathbf{y}}_{\mathrm{B}}$ . Then, the MSE matrix at Bob can be written as

$$\begin{array}{rl}\mathbf{E}& ={\mathbb{E}}_{\mathbf{s},{\mathbf{n}}_{\mathrm{B}}}\left[(\tilde{\mathbf{s}}-\mathbf{s})(\tilde{\mathbf{s}}-\mathbf{s}{)}^{\mathrm{H}}\right]\\ =& \left({\mathbf{I}}_{M_{\mathrm{R}}}-{\mathrm{U}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}\right){\left({\mathbf{I}}_{M_{\mathrm{R}}}-{\mathrm{U}}^{\mathrm{H}}{\mathrm{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}\right)}^{\mathrm{H}}+{\sigma }_{\mathrm{R}}^2{\mathbf{U}}^{\mathrm{H}}\mathbf{U}.\end{array}\text{\hspace{1em}(27)}$$

By introducing a weight matrix $\Psi \succcurlyeq 0$ for Bob, the subproblem (26) can be equivalently reformulated as [38 , Thm. 1]

$$\begin{array}{ll}{\min }_{\mathbf{\Psi },\mathbf{U},{\mathbf{W}}_{\mathrm{FD}}}& \mathrm{Tr}(\mathbf{\Psi })-{\log }_2|\mathbf{\Psi }|\\ \text{ s.t. }& (26b),(26\mathrm{c}).\end{array}\text{\hspace{1em}(28)}$$

Although the transformed problem has more optimization variables than (26), the objective function in (28) is more tractable. The receive beamforming matrix $\mathbf{U}$ and the weight matrix $\Psi$ only appear in the objective function (28a). By setting the derivatives of (28a) with respective to $\mathbf{U}$ and $\Psi$ to zero, respectively, the optimal solutions can be obtained as

$$\begin{array}{l}{\mathbf{U}}^{\star }={\left({\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}{\mathbf{W}}_{\mathrm{FD}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}^{\mathrm{H}}+{\sigma }_{\mathrm{B}}^2{\mathbf{I}}_{M_{\mathrm{B}}}\right)}^{-1}{\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}},\\ {\mathbf{\Psi }}^{\star }={\mathbf{E}}^{-1}\end{array}\text{\hspace{1em}(29)}$$

Substituting the optimal \mathbf{U}^{\star} in (29) into (27) yields the optimal MSE matrix as follows

$$\begin{array}{l}{\mathbf{E}}^{\star }=\\ {\mathbf{I}}_{M_{\mathrm{B}}}-{\mathbf{W}}_{\mathrm{FD}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}^{\mathrm{H}}{\left({\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}{\mathbf{W}}_{\mathrm{FD}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}^{\mathrm{H}}+{\sigma }_{\mathrm{B}}^2{\mathbf{I}}_{M_{\mathrm{B}}}\right)}^{-1}{\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}.\end{array}\text{\hspace{1em}(30)}$$

Substituting (27) into the objective function of (28) and discarding the constant terms, the problem that updates the full-digital beamforming matrix \mathbf{W}_{\mathrm{FD}} is transformed as

$$\begin{array}{ll}{\min }_{{\mathbf{W}}_{\mathrm{FD}}}& \mathrm{Tr}\left({\mathbf{W}}_{\mathrm{FD}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}^{\mathrm{H}}\mathbf{U}\mathbf{\Psi }{\mathbf{U}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}\right)-\mathrm{Tr}\left(\mathbf{\Psi }{\mathbf{W}}_{\mathrm{FD}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}^{\mathrm{H}}\mathbf{U}\right)\\ & -\mathrm{Tr}\left(\mathbf{\Psi }{\mathbf{U}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{B}}{\mathbf{W}}_{\mathrm{FD}}\right)\\ \text{ s.t. }& (26\mathrm{b}),(26\mathrm{c}).\end{array}$$

WMMSE与BCD构造的对比辨析

WMMSE变换后的表达式(27)与$\mathbf{E}(\mathbf{U},\mathbf{V})$极为相似，

出处

https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7018097 lemma 4.1

原文地址：https://blog.csdn.net/qq_45542321/article/details/136234652

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