## 3 Proposed Algorithm
For \(a\geq 1\), let \(\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{a}>0\) denote the nonzero eigenvalues of \(\mathbf{A}\). It follows

\[|\mathbf{I}+\mathbf{A}| =\prod_{i=1}^{a}(1+\lambda_{i})\] \[=1+\prod_{i=1}^{a}\lambda_{i}+\prod_{i\
eq k}^{a}\lambda_{i} \lambda_{k}+...\] \[\geq 1+\prod_{i=1}^{a}\lambda_{i}=1+\text{tr}(\mathbf{A})\]

and it can be seen that the inequality above holds if and only if \(a=1\). ## Appendix B

**Proof of Theorem 1. ** Since problem (31) is convex, the optimal solution \(\Omega^{(n)}\) is obtained by solving (31) for a given \(\Omega^{(n-1)}\). Due to the solution \(\Omega^{(n-1)}\) obtained in the \(n\)-th iteration, \(\Omega^{(n)}\) can be updated in the \((n+1)\)-th iteration. Based on (28) and (29), we can obtain the following inequalities in the \((n+1)\)-th iteration. \[ \begin{split}&\sum_{j=1}^{K}\left(\mathbf{w}_{j}{}^{H}\mathbf{H}_{c,k }\mathbf{w}_{j}\right)+\mathbf{z}^{H}\mathbf{H}_{c,k}\mathbf{z}+\sum_{m=1}^{M} \left(\mathbf{v}_{m}^{H}\mathbf{G}_{c,k}\mathbf{v}_{m}\right)\\ \geq&\sum_{j=1}^{K}\left(\mathbf{\tilde{w}}_{j}{}^{*H }\mathbf{H}_{c,k}\mathbf{\tilde{w}}_{j}+2\Re\{\mathbf{\tilde{w}}_{j}^{*H} \mathbf{H}_{c,k}\Delta\mathbf{w}_{j}\}\right)+\mathbf{\tilde{z}}{}^{*H} \mathbf{H}_{c,k}\mathbf{\tilde{z}}+2\Re\{\mathbf{\tilde{z}}{}^{*H}\mathbf{H}_ {c,k}\Delta\mathbf{z}\}\\ &\quad+\sum_{m=1}^{M}(\mathbf{\tilde{v}}_{m}^{*H}\mathbf{G}_{c,k }\mathbf{\tilde{v}}_{m}+2\Re\{\mathbf{\tilde{v}}_{m}^{*H}\mathbf{G}_{c,k} \Delta\mathbf{v}_{m}\})\\ \geq&\frac{\tilde{E}^{c}}{\eta_{k}(1-\rho_{k})}- \sigma_{a,k}^{2}\
u_{k},\\ &\quad\sum_{j=1}^{K}\left(\mathbf{w}_{j}{}^{H}\mathbf{\tilde{H}} _{c,j}\mathbf{w}_{j}\right)+\mathbf{z}^{H}\mathbf{\tilde{H}}_{c,j}\mathbf{z}+ \sum_{m=1}^{M}(\mathbf{v}_{m}{}^{H}\mathbf{\tilde{G}}_{c,l}\mathbf{v}_{m})\\ \geq&\sum_{j=1}^{K}\left(\mathbf{\tilde{w}}_{j}{}^{ *H}\mathbf{\tilde{H}}_{c,l}\mathbf{\tilde{w}}_{j}+2\Re\{\mathbf{\tilde{w}}_{j} ^{*H}\mathbf{\tilde{H}}_{l}\Delta\mathbf{w}_{j}\}\right)+\mathbf{\tilde{z}}{}^{* H}\mathbf{\tilde{H}}_{c,l}\mathbf{\tilde{z}}+2\Re\{\mathbf{\tilde{z}}{}^{*H} \mathbf{\tilde{H}}_{c,l}\Delta\mathbf{z}\}\\ &\quad+\sum_{m=1}^{M}(\mathbf{\tilde{v}}_{m}^{*H}\mathbf{\tilde{ G}}_{c,l}\mathbf{\tilde{v}}_{m}+2\Re\{\mathbf{\tilde{v}}_{m}^{*H}\mathbf{ \tilde{G}}_{c,l}\Delta\mathbf{v}_{m}\})\\ \geq&\frac{\tilde{E}^{c}}{\eta_{l}}-N_{E}\sigma_{c,l }^{2},\
u_{l},\end{split}  \tag{24}\]

which are conservative convex approximations with \(\mathbf{w}_{j}=\mathbf{\tilde{w}}_{j}^{*}+\Delta\mathbf{w}_{j}\), \(\mathbf{z}=\mathbf{\tilde{z}}^{*}+\Delta\mathbf{z}\) and \(\mathbf{v}_{m}=\mathbf{\tilde{v}}_{m}^{*}+\Delta\mathbf{v}_{m}\). According to the linear approximations employed in (A.1) and (A.2), the rule for updating in Algorithm 1 ensures that \(\{\mathbf{w}_{j}^{*}\}\), \(\{\mathbf{z}^{*}\}\) and \(\{\mathbf{v}_{m}^{*}\}\) has always been feasible to problem (31) at the \((q+1)\)-th iteration. Thus, we immediately have \(f_{n+1}^{*}(\gamma_{c}^{(n+1)},\gamma_{c}^{(n+1)})\geq f_{n}^{*}(\gamma_{c}^{ (n)},\gamma_{c}^{(n)})\). In other words, this behaviour of \(\Omega^{(n)}\) reveals that the proposed algorithm has always a monotonically non-decreasingsequence of the objective values. On account of the transmit power constraint at the macrocell sink node and femtocell sink node, problem (31) has an upper bound on the MSC. Therefore, the proposed algorithm can be guaranteed to converge to some optimum solution, i.e., \(\left\|\Delta\mathbf{w}_{j}\right\|=0\), \(\left\|\Delta\mathbf{z}\right\|=0\), \(\left\|\Delta\mathbf{v}_{m}\right\|=0\), and \(f_{n+1}^{*}(\gamma_{c}^{(n+1)},\gamma_{e}^{(n+1)})=f_{n}^{*}(\gamma_{c}^{(n)},\gamma_{e}^{(n)})\). ## References

* (1) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Towards smart factory for industry 4.0: A self-organized multi-agent system with big data based feedback and coordination. _Comput. Netw._**2016**, _101_, 158-168.