## 3 Proposed Algorithm  
In this section, we will propose an iterative algorithm to obtain near-optimal solution by applying the SCA method and SPCA technology [41]. To tackle problem (8) we firstly introduce an equivalent reformulation about problem (8). To circumvent this issue, problem (8) can be rewritten as  
\[\max_{\mathbf{w}_{k},\mathbf{z},\mathbf{v}_{m},\rho_{k}}\ \min_{m=1,\ldots,M}\ \log\left(1+\frac{\mathbf{g}_{m}^{H}\mathbf{v}_{m}\mathbf{v}_{m}^{H}\mathbf{g }_{m}}{\mathbf{h}_{m}^{H}\big{(}\sum_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H} \mathbf{+z}\mathbf{z}^{H}\big{)}\mathbf{h}_{m}+\mathbf{g}_{m}^{H}\sum_{j\
eq m }(\mathbf{v}_{j}\mathbf{v}_{j}^{H})\mathbf{g}_{m}\mathbf{+}\sigma_{m}^{2}}\right)\] \[\qquad-\log\left|\mathbf{I}+(\mathbf{H}_{e,l}^{H}\big{(}\sum_{k=1 }^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{+z}\mathbf{z}^{H}\big{)}\mathbf{ H}_{e,l}+\mathbf{G}_{e,l}^{H}\sum_{j\
eq m}(\mathbf{v}_{j}\mathbf{v}_{j}^{H}) \mathbf{G}_{e,l}\mathbf{+}\sigma_{e,l}^{2}\mathbf{I})^{-1}\mathbf{G}_{e,l}^{H} \mathbf{v}_{m}\mathbf{v}_{m}^{H}\mathbf{G}_{e,l}\right|\] (9a) \[s.t.\ \log\left(1+\frac{\rho_{k}\mathbf{h}_{e,k}^{H}\mathbf{w}_{k} \mathbf{w}_{k}^{H}\mathbf{h}_{c,k}}{\rho_{k}\big{(}\mathbf{h}_{c,k}^{H}(\sum_ {j\
eq k}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{+z}\mathbf{z}^{H})\mathbf{h} _{c,k}\mathbf{+}\mathbf{g}_{c,k}^{H}\sum_{m=1}^{M}(\mathbf{v}_{m}\mathbf{v}_{ m}^{H})\mathbf{g}_{c,k}\mathbf{+}\sigma_{a,k}^{2}\big{)}\mathbf{+}\delta_{p,k}^{2}}\right)\] \[\qquad-\log\left|\mathbf{I}+\big{(}\sigma_{e,l}^{2}\mathbf{I}+ \mathbf{H}_{e,l}^{H}\big{(}\sum_{j\
eq k}\mathbf{w}_{j}\mathbf{w}_{j}^{H} \mathbf{+z}\mathbf{z}^{H}\big{)}\mathbf{H}_{e,l}\mathbf{+}\mathbf{G}_{e,l}^{ H}\sum_{m=1}^{M}\big{(}\mathbf{v}_{m}\mathbf{v}_{m}^{H})\mathbf{G}_{e,l}\big{)}^{-1} \mathbf{H}_{e,l}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{H}_{e,l}\right| \geq\bar{R}^{c},\] (9b) \[\eta_{k}(1-\rho_{k})\big{(}\mathbf{r}\big{(}\mathbf{h}_{c,k}^{H} (\sum_{j=1}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{+z}\mathbf{z}^{H}) \mathbf{h}_{c,k}\big{)}+\text{tr}\big{(}\mathbf{g}_{c,k}^{H}\sum_{m=1}^{M} \big{(}\mathbf{v}_{m}\mathbf{v}_{m}^{H})\mathbf{g}_{c,k}\big{)}+N_{C}\sigma_{a, k}^{2}\big{)}\geq\bar{E}^{c},\forall k,\] (9c) \[\eta_{l}\big{(}\mathbf{r}\big{(}\mathbf{H}_{e,l}^{H}\big{(}\sum_{ j=1}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{+z}\mathbf{z}^{H}\big{)} \mathbf{H}_{e,l}\big{)}+\text{tr}\big{(}\mathbf{G}_{e,l}^{H}\sum_{m=1}^{M} \big{(}\mathbf{v}_{m}\mathbf{v}_{m}^{H})\mathbf{G}_{e,l}\big{)}+\sigma_{e,l}^{ 2}\big{)}\geq\bar{E}^{e},\forall l,\] (9d) (8d), (8e). It is observed that the determinant functions is in the \(R_{c,lk}\) of the objective function (9a). We consider the following Lemma 1, which will be applied to provide a convex approximation form to \(C_{l,k}\). **Lemma 1**. : _([42]) For any matrix \(\mathbf{A}\), which \(\mathbf{A}\) is positive semi-definite, i.e., \(\mathbf{A}\succeq\mathbf{0}\). It holds true that_  
\[|\mathbf{I}+\mathbf{A}|\geq 1+\text{tr}(\mathbf{A}), \tag{10}\]  
_where the equality holds if and only if \(\text{rank}(\mathbf{A})\leq 1\)._  
Proof. : See Appendix A. We introduce two real-valued slack variables \(\gamma_{c}\) and \(\gamma_{e}\). By applying the Lemma 1, problem (9) can be rewritten as  
In addition, for the constraint (9b), we will reformulate it by using SPCA method.