## 3 The Cooperative Beam Association and Power Allocation Scheme
### SCA-Based Power Allocation  
By solving the beam association problem, we can reformulate the optimization problem as **P2**. \[\textbf{P2}:\min_{\{P_{k,m}\},\forall k,\forall m.}\max\ (\Phi_{m}+\sum_{k=1}^{K} \sum_{m=1}^{M}P_{k,m}) \tag{9}\]  
s.t. \[\sum_{k=1}^{K}BW\cdot\log_{2}(1+R_{k,m})\leq T_{m}\] (9a) \[(7c),(7e),(7f)\] (9b)  
To simplify the optimization target, an auxiliary variable \(\lambda\) is introduced to replace the \(\Phi_{m}+\sum_{k=1}^{K}\sum_{m=1}^{M}P_{k,m}\). Take (1), (9a), and (9b) into the optimization problem, **P2** can be further expressed as **P3**. \[\textbf{P3}:\min_{\{P_{k,m}\},\forall k,\forall m,\lambda}\lambda \tag{10}\]  
s.t. \[\sum_{k=1}^{K}log_{2}(1+\frac{v_{k,m}P_{k,m}G_{tx}h_{k,m}G_{rx}P_{ rx}}{I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW})\leq T_{m}\] (10a) \[T_{m}-\sum_{k=1}^{K}log_{2}(1+\frac{v_{k,m}P_{k,m}G_{tx}h_{k,m}G _{rx}P_{rx}}{I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW})+\sum_{k=1}^{K}\sum_{m=1}^{M} P_{k,m}\leq\lambda\] (10b) \[(7e),(7f)\] (10c)  
The optimization problem **P3** is still a non-convex problem due to non-convex constraints (10a) and (10b). To tackle it, the SCA method is adopted. Since \(I_{L}(k,m)\) as (3) contains \(P_{k,m}\), we cannot directly make a first-order Tayor expansion for (10a) and (10b). We first split the fractional form in the logarithmic function into the different operations of two logarithmic functions. \[ \begin{array}{l}\sum_{k=1}^{K}\log_{2}\left(1+\frac{v_{k,m}P_{k,m}G_{tx}h_{k, m}G_{rx}P_{rx}}{I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW}\right)\leq T_{m}\\ \Leftrightarrow\sum_{k=1}^{K}\left[\log_{2}(v_{k,m}P_{k,m}G_{tx}h_{k,m}G_{rx}P _{rx}+I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW)\right.\\ \left.-\log_{2}(I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW))\right]\leq T_{m}\end{array}  \tag{11}\]  
In (11), \(-\log_{2}(I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW)\) is convex, let \(T_{k,m}=I_{G}(k,m)+N_{0}\cdot BW\), and we take a first-order Tayor expansion for the first part of (11) as (12). \[ \begin{array}{l}\log_{2}\left(\sum_{i=1}^{K}\sum_{j=1,j\
eq m}^{M}v_{ij}P_{ ij}G_{tx}\left(\hat{x}_{ij}^{k,m}\right)h_{ij}G_{rx}\left(\hat{p}_{ij}^{k,m} \right)P_{rx}+T_{k,m}+v_{k,m}P_{k,m}G_{tx}H_{k,m}G_{rx}P_{rx}\right)\\ \approx\log_{2}\left(\sum_{i=1}^{K}\sum_{j=1,j\
eq m}^{M}v_{ij}P_{ij}^{t}G_{tx} \left(\hat{x}_{ij}^{k,m}\right)h_{ij}G_{rx}\left(\hat{p}_{ij}^{k,m}\right)P_{ rx}+T_{k,m}+v_{k,m}P_{k,m}^{(t)}G_{tx}h_{k,m}G_{rx}P_{rx}\right)\\ +\frac{\sum_{i=1}^{K}\sum_{j=1,j\
eq m}^{M}v_{ij}G_{tx}\left(\hat{x}_{ij}^{k,m} \right)h_{ij}G_{rx}\left(\hat{p}_{ij}^{k,m}\right)P_{rx}\left(\hat{p}_{ij}^{k,m}\right)P_{rx}\left(\hat{p}_{ij}^{k,m}\right)+v_{k,m}G_{tx}h_{k,m}G_{rx}P_{rx} \left(\hat{p}_{rx}-P_{k,m}^{(t)}\right)}{\left(\sum_{i=1}^{K}\sum_{j=1,j\
eq m }^{M}v_{ij}P_{ij}^{t}G_{tx}\left(\hat{x}_{ij}^{k,m}\right)h_{ij}G_{rx}\left( \hat{p}_{ij}^{k,m}\right)P_{rx}+T_{k,m}+v_{k,m}P_{k,m}^{(t)}G_{tx}h_{k,m}G_{rx} P_{rx}\right)\cdot\ln 2}=V_{1}(k,m)\end{array}  \tag{12}\]  
Then, constraint (11) can be written as (13). With the same operation, we can approximate (10b) as (14) with SCA. \[\sum_{i=1}^{K}[V_{1}(k,m)-\log_{2}(I_{L}(k,m)+I_{G}(k,m)+N_{0}\cdot BW)]\leq T _{m} \tag{13}\]  
\[T_{m}-\sum_{k=1}^{K}\log_{2}(I_{L}(k,m)+T_{k,m}+v_{k,m}P_{k,m}G_{tx}h_{k,m}G_{ rx}P_{rx})-V_{2}(k,m))+\sum_{k=1}^{K}\sum_{m=1}^{M}P_{k,m}\leq\lambda \tag{14}\]where  
\[ \begin{array}{l}V_{2}(k,m)=\log_{2}\left(\sum_{i=1}^{K}\sum_{j=1,i\
eq n}^{M}v_{ ij}G_{\mathrm{tx}}\left(a_{i,j}^{k,m}\right)P_{i,j}^{(l)}h_{ij}G_{\mathrm{ix}} \left(\beta_{i,j}^{k,m}\right)P_{rx}+T_{k,m}\right)\\ +\frac{\sum_{i=1}^{K}\sum_{j=1,i\
eq n}^{M}v_{ij}G_{\mathrm{ix}}\left(a_{i,j}^{ k,m}\right)h_{ij}G_{\mathrm{ix}}\left(\beta_{i,j}^{k,m}\right)P_{rx}\left(P_{ij,j}-p_{i,j}^{(l)}\right)}{\left(\sum_{i=1}^{K}\sum_{j=1,i\
eq n}^{M}v_{ij}G_{ \mathrm{ix}}\left(a_{i,j}^{k,m}\right)P_{i,j}^{(l)}h_{ij}G_{\mathrm{ix}}\left( \beta_{i,j}^{k,m}\right)P_{rx}+T_{k,m}\right)\ln 2}\end{array} . \tag{15}\]  
Then, the approximated convex problem **P4** is formed by replacing (10a), (10b) as (13), (14), which can be optimally solved by the CVX toolbox.