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

出处

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