Algorithm 1 Energy-efficient beamforming and power allocation.

1: Initialize ${\overrightarrow{p}}^{\left(0\right)}$, $\overrightarrow{\lambda }$, and $\overrightarrow{\mu }$, randomly 2: Determine ${\mathbf{w}}_i=\frac{{\mathbf{g}}_{i,i}}{\parallel {\mathbf{g}}_{i,i}\parallel }$, $\forall i\in \mathbb{N}$ 3: repeat 4:     Set $\overrightarrow{q}=\overrightarrow{r}/{\overrightarrow{P}}^{CE}$ 5:     $j\leftarrow 1$ 6:     repeat 7:         ${\overrightarrow{p}}_{\mathrm{old}}\leftarrow {\overrightarrow{p}}^{(j-1)}$ 8:         for $i=1$ to N 9:             Compute $p_i^{\left(j\right)}$ according to (16) 10:             Update ${\lambda }_i$ and ${\mu }_i$ according to (17) and (18) 11:         end for 12:         ${\overrightarrow{p}}^{\left(j\right)}=\{p_1^{\left(j\right)},p_2^{\left(j\right)},\cdots ,p_N^{\left(j\right)}\}$ 13:         $j\leftarrow j+1$ 14:     until $\parallel {\overrightarrow{p}}^{\left(j\right)}-{\overrightarrow{p}}_{\mathrm{old}}\parallel <ϵ$ 15:     Update $\overrightarrow{r}$ and ${\overrightarrow{P}}^{CE}$ with $\overrightarrow{p}$ 16: until$\parallel \overrightarrow{r}-\overrightarrow{q}{\overrightarrow{P}}^{CE}\parallel <ϵ$