## 3 Solution of the Optimization Problem
### Optimality Conditions for SDP Relaxation
\begin{split}&\operatorname{rank}(\mathbf{R}^{*})=\operatorname{rank}(\mathbf{T}^ {*}+(\beta^{*}+\gamma^{*})\mathbf{H}),\\ &\geq\operatorname{rank}(\mathbf{T}^{*})-\operatorname{rank}((\beta^{* }+\gamma^{*})\mathbf{H})=N_{T}-1.\end{split}  \tag{38}\]However, if \(\text{rank}(\mathbf{R}^{*})=N_{T}\), then according to (33) and (34) it follows that \(\mathbf{W}^{*}=0\), which, of course, cannot be the optimal solution to optimization problem (20). Thus, we obtain \(\text{rank}(\mathbf{T}^{*})=N_{T}-1\) and get the optimal solution \(\mathbf{W}^{*}=\mathbf{a}\mathbf{t}^{H}\), \(a\geq 0\) if \(\tilde{\mathbf{r}}=N_{T}\) where \(\mathbf{\iota}\) lies in the null space of \(\mathbf{R}^{*}\) with unit norm. Next, we consider the cast where \(\mathbf{T}^{*}\) is rank defective, i.e., \(\tilde{\mathbf{r}}\leq N_{T}\). In this case, the matrix \(\mathbf{N}_{1}\) is modeled as the standard orthogonal basis that spans the null space of \(\mathbf{T}^{*}\), i.e., \(\mathbf{T}^{*}\mathbf{N}_{1}=\mathbf{0}\). Let \(\pi_{1,n}\) represent the \(n\)-th column of \(\mathbf{N}_{1}\), \(1\leq n\leq N_{T}-\tilde{\mathbf{r}}\). The following equation holds:  
\[ \begin{split}&\pi_{1,n}^{H}\mathbf{R}^{*}\mathbf{\pi}_{1,n}=\mathbf{\pi}_{1,n }^{H}\left(\mathbf{T}^{*}+\left(\beta^{*}+\gamma^{*}\right)\mathbf{H}\right)\mathbf{\pi}_{ 1,n}\\ &=\left(\beta^{*}+\gamma^{*}\right)\pi_{1,n}^{H}\mathbf{H}\mathbf{\pi}_{1,n}=\left(\beta^{*}+\gamma^{*}\right)\pi_{1,n}^{H}\mathbf{h}\mathbf{h}^{H}\mathbf{\pi}_{1,n}.\end{split}  \tag{39}\]  
Since \(\mathbf{R}^{*}\preceq\mathbf{0}\) and \(|\mathbf{\pi}_{1,n}^{H}\mathbf{n}|\geq 0\), it follows that  
\[\mathbf{R}^{*}\mathbf{N}_{1}=\mathbf{0}\ \ \ and\ \ \ \mathbf{H}\mathbf{N}_{1}=\mathbf{0}. \tag{40}\]  
Furthermore, according to (37), another inequality is achieved, i.e., \(\text{rank}(\mathbf{R}^{*})\geq\text{rank}(\mathbf{T}^{*})-\text{rank}(\mathbf{H})=\tilde{ \mathbf{r}}-1\). We define \(\mathbf{\Omega}_{1}\) as the orthogonal basis for the null space of \(\mathbf{R}^{*}\); it then yields  
\[\text{rank}(\mathbf{\Omega}_{1})=N_{T}-\text{rank}(\mathbf{R}^{*})\leq N_{T}-\tilde{ \mathbf{r}}+1. \tag{41}\]  
Next, we come to show that \(\text{rank}(\mathbf{\Omega}_{1})=N_{T}-\tilde{\mathbf{r}}+1\). With (40), \(\mathbf{N}_{1}\) spans \(N_{T}-\tilde{\mathbf{r}}\) orthogonal dimensions of the null space of \(\mathbf{A}^{*}\), i.e., \(\text{rank}(\mathbf{\Omega}_{1})\geq N_{T}-\tilde{\mathbf{r}}\). Assume that \(\text{rank}(\mathbf{\Omega}_{1})=N_{T}-\tilde{\mathbf{r}}\), then, we get \(\mathbf{\Omega}_{1}=\mathbf{N}_{1}\) and express \(\mathbf{W}^{*}\) as \(\mathbf{W}^{*}=\sum_{n=1}^{N_{T}-\tilde{\mathbf{r}}}b_{n}\pi_{1,n}\pi_{1,n}^{H}\), where \(b_{n}\geq 0\), \(\forall n\). Moreover, in this case, there is no information transferred to information receiver according to (40). Therefore, according to (41), there exists only one single subspace spanned by unit norm vector \(\mathbf{\iota}\) that also satisfies \(\mathbf{N}_{1}\mathbf{\iota}=\mathbf{0}\). Therefore, we have  
\[\mathbf{\Omega}_{1}= \begin{bmatrix}\mathbf{N}_{1}\ \ \mathbf{\iota}\end{bmatrix}  \tag{42}\]  
and \(\text{rank}(\mathbf{\Omega}_{1})=N_{T}-1+\tilde{\mathbf{r}}\). Furthermore, according to (33) and (34), any optimal solution \(\mathbf{W}^{*}\) for optimization problem (20) can be modeled as  
\[\mathbf{W}^{*}=\sum_{n=1}^{N_{T}-\tilde{\mathbf{r}}}b_{n}\mathbf{\pi}_{1,n}\mathbf{\pi}_{1,n}^{ H}+\mathbf{a}\mathbf{\iota}^{H}, \tag{43}\]  
where \(b_{n}\geq 0\), \(\forall n\), and \(a\geq 0\). The first part of Theorem 1 is thus proved. Next, we prove the second part of Theorem 1. Suppose we obtained the optimal solution \(\left\{\mathbf{W}^{*},\mathbf{V}^{*},\mathbf{V}^{*}_{0},\rho^{*},t^{*}\right\}\), where \(\mathbf{W}^{*}\) is given by (23) and \(\text{rank}(\mathbf{W}^{*})>1\). Then, the new alternative solution \(\left\{\tilde{\mathbf{W}}^{*},\tilde{\mathbf{V}}^{*},\tilde{\mathbf{V}}^{*}_{0},\tilde{ \rho}^{*},\tilde{\mathbf{r}}^{*}\right\}\) can be given in (23)-(26) and has the following properties:  
\[