## 3 Proposed Algorithm
\begin{split}&\sum_{j=1}^{K}\left(\mathbf{\tilde{w}}_{j}^{H} \mathbf{\tilde{H}}_{e,l}\mathbf{\tilde{w}}_{j}+2\Re\{\mathbf{\tilde{w}}_{j}^{H }\mathbf{\tilde{H}}_{l}\Delta\mathbf{w}_{j}\}\right)+\mathbf{\tilde{z}}^{H} \mathbf{\tilde{H}}_{e,l}\mathbf{\tilde{z}}+2\Re\{\mathbf{\tilde{z}}^{H} \mathbf{\tilde{H}}_{e,l}\Delta\mathbf{z}\}\\ &\quad+\sum_{m=1}^{M}(\mathbf{\tilde{v}}_{m}^{H}\mathbf{\tilde{G} }_{e,l}\mathbf{\tilde{v}}_{m}+2\Re\{\mathbf{\tilde{v}}_{m}^{H}\mathbf{\tilde {G}}_{e,l}\Delta\mathbf{v}_{m}\})\geq\frac{\tilde{E}^{c}}{\eta_{l}}-N_{E} \sigma_{e,l}^{2},\forall l.\end{split}  \tag{29}\]  
Finally, by rearranging (8d) as  
\[ \begin{split}&\|[\mathbf{w}_{1}^{T},...,\mathbf{w}_{K}^{T}, \mathbf{z}^{T},\mathbf{v}_{1}^{T},...,\mathbf{v}_{M}^{T}]\|\leq\sqrt{P},\\ &\|[\mathbf{w}_{1}^{T},...,\mathbf{w}_{K}^{T}]\|\leq\sqrt{P_{1}},\\ &\|[\mathbf{v}_{1}^{T},...,\mathbf{v}_{M}^{T}]\|\leq\sqrt{P_{2}}.\end{split}  \tag{30}\]  
Eventually, the original MSCM problem (8) is transformed into following convex form as  
\[\begin{split}&\max_{\mathbf{w}_{k},\mathbf{z},\mathbf{v}_{m}, \
eq\tilde{w}_{k},\gamma_{e},\gamma_{e},\omega_{m},\omega_{m},\omega_{m},t_{l}, \sigma_{l},s_{1},s_{2}}\\ &\text{s.t. \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:mscsc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:msc}, \eqref{eq:mscsc}, \eqref{eq:msc}, \eqref{eq:```
1. Given the tolerance of accuracy \(\phi\) (very small value), and the maximum iteration number \(N^{\max}\). 2. Set \(n=0\) and initialize \(\Omega^{(0)}=\{\tilde{\mathbf{w}}_{k}^{(0)},\tilde{\mathbf{w}}_{m}^{(0)},\tilde{ \mathbf{z}}^{(0)},\tilde{w}_{m}^{(0)},\tilde{u}_{m}^{(0)},\tilde{t}_{l}^{(0)}, \gamma_{e}^{(0)},\tilde{s}_{1}^{(n)},\tilde{s}_{2}^{(n)}\}\). 3. For\(n=1:N^{\max}\)do
4. Solve problem (31) with \(\Omega^{(n)}=\left\{\tilde{\mathbf{w}}_{k}^{(n)},\tilde{\mathbf{w}}_{m}^{(n)},\tilde{\mathbf{z}}^{(n)},\tilde{w}_{m}^{(n)},\tilde{u}_{m}^{(n)},\tilde{t}_{l }^{(n)},\tilde{\gamma}_{e}^{(n)},\tilde{s}_{1}^{(n)},\tilde{s}_{2}^{(n)}\right\}\) and denote the solution \(\Omega\) by \(\Omega^{*}\). 5. Set \(\Omega^{(n+1)}=\Omega^{*}\). 6. Update the iteration number \(n\gets n+1\). 7. The objective function converges, i.e., \(\left|\left(\gamma_{e}^{(n+1)}-\gamma_{e}^{(n+1)}\right)-\left(\gamma_{c}^{( n)}-\gamma_{e}^{(n)}\right)\right|\leq\phi\), or the maximum number of iterations is reached, i.e., \(n\geq N^{\max}\). 8. endfor
9. Obtain \(\{\mathbf{w}_{k}\}_{k=1}^{K}\), \(\{\mathbf{v}_{m}\}_{m=1}^{M}\) and \(\mathbf{z}\). ```

**Algorithm 1** Proposed SCA-SPCA-based iterative Algorithm

Theorem 1: _Suppose that \(\gamma_{c}^{(n)}\) and \(\gamma_{e}^{(n)}\) are the optimal solution to problem (31) at the \(n\)-th iteration. Denoting \(f_{n}^{*}(\gamma_{c}^{(n)},\gamma_{e}^{(n)})\) as the optimal objective function at the \(q\)-th iteration, we have \(f_{n+1}^{*}(\gamma_{c}^{(n+1)},\gamma_{e}^{(n+1)})\geq f_{n}^{*}(\gamma_{c}^{ (n)},\gamma_{e}^{(n)})\). Thus, with the increase of \(n\), \(\|\Delta\mathbf{w}_{k}\|\), \(\|\Delta\mathbf{z}\|\) and \(\|\Delta\mathbf{v}_{m}\|\) approach 0. Hence, Algorithm 1 will converge._

Proof: See Appendix B.