## 3 Joint Optimization of EE-SE
### Quadratic Transform
#### 3.2.2 Optimization of Phase-Shifting Values for Given Transmitting Precoding  
To optimize problem (P6) over \(\mathbf{\Theta}\), the terms which do not correlate with \(\mathbf{\Theta}\) can be omitted, and new problem forms as  
\[ \begin{split}\text{(P8)}&\min_{\mathbf{\Theta}}\,\sum_{k =1}^{K}\omega_{k}\alpha_{k}\epsilon_{k}\\ &\text{s.t. \eqref{eq:P8}}.\end{split}  \tag{18}\]  
As variable \(\mathbf{\Theta}\) is a diagonal matrix, we can transform the variable of problem (P8) into vector \(\mathbf{\theta}\) through some substitution and operation and rewrite the objective function in problem (P8) in the quadratic form of \(\mathbf{\theta}\). Therefore, we add some auxiliary variables \(\mathbf{a}_{i,k}=\sqrt{\eta}\text{diag}(\mathbf{h}_{k,k}^{H})\mathbf{G}^{H} \mathbf{w}_{i}\) and \(b_{i,k}=\mathbf{h}_{\text{d},k}^{H}\mathbf{w}_{i}\). Thus, we have \(\Big{(}\mathbf{h}_{\text{d},k}^{H}+\mathbf{h}_{t,k}^{H}\mathbf{\Theta}^{H} \mathbf{G}^{H}\Big{)}\mathbf{w}_{i}=b_{i,k}+\mathbf{\theta}^{H}\mathbf{a}_{i,k}\), where \(\mathbf{\theta}=[\theta_{1},\dots,\theta_{N}]^{T}\)[26]. Then, we can transform \(e_{k}\) into the new form below to simplify the expression of the optimization problem. Therefore, we can rewrite the objective function in (18), and problem (P8) can be equivalently expressed as new problem (P8) below. \[\begin{split} e_{k}&=|u_{k}^{*}(\mathbf{b}_{k,k}+ \mathbf{\theta}^{H}\mathbf{a}_{k,k})-1|^{2}+\sum_{i=1,i\
eq k}^{K}\left|u_{i}( \mathbf{b}_{i,k}+\mathbf{\theta}^{H}\mathbf{a}_{i,k})\right|^{2}+\sigma_{0}^ {2}|u_{k}|^{2}\\ &=\mathbf{\theta}^{H}\Bigg{(}\sum_{i=1}^{K}|u_{i}|^{2}\mathbf{a} _  
The average convergence performance of the proposed algorithm is illustrated in Figure 2. The result demonstrates that the convergence is achieved with a small number of iterations. The EE-SE trade-off curves attained by the algorithm framework under different values of BS power constraint \(P_{\text{T}}\) and weight \(a\) are shown in Figure 3. Obviously, EE-SE trade-offs under different weighting factors are similar to each other when the value of \(P_{\text{T}}\) is small, approximately when \(P_{\text{T}}<20\) dBm, as all available power of BS can be fully utilized and both EE and SE are maximized when \(P_{\text{T}}\) is small. Therefore, changes in values of \(a\) have little impact on the EE-SE trade-off when \(P_{\text{T}}\) is small. When \(P_{\text{T}}\) gets high, the EE-SE trade-offs under different \(a\) differ greatly. It is observed that the increasing of \(a\) brings about an increase in EE optimization and a decrease in SE, and the decreasing of \(a\) causes a decrease in EE and an increase in SE. That is because larger weight is given to EE for larger \(a\), and thus more focus is transferred to EE when conducting the optimization. However, more focus is transferred to the SE optimization when a smaller weight is given to EE for smaller \(a\). \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Values** \\ \hline Path loss & \(-110\) dB \\ Number of users \(K\) & \(4\) \\ Weight for each user in the system’s SE & \(\frac{1}{K}\) \\ Number of RIS reflecting units \(N\) & \(8\) \\ Number of BS antennas \(M\) & \(4\) \\ The reciprocal of the power amplifier efficiency at the BS \(\mu\) & \(5\) \\ The reflection efficiency of the RIS \(\eta\) & \(0.8\) \\ The noise variance at the users \(\sigma_{0}^{2}\) & \(-130\) dBm \\ Static power consumption of each user \(P_{k}\) & \(10\) dBm \\ Hardware dissipated power at the BS \(P_{\text{BS}}\) & \(40\) dBm \\ Per-element static power at the RIS \(P_{\text{n}}\) & \(20\) dBm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation setup parameters. Figure 2: Average convergence performances versus the number of iterations for \(a=0.5\). In Figure 4, we show the EE-SE trade-offs of the system under the condition of \(a=0.5\).