# Phase Shift Design in RIS Empowered Wireless Networks: From Optimization to AI-Based Methods
## 3 Review on Optimization Methods under Continuous Phase Shift
### Summary and Performance Comparison  
To summmarize, the optimization methods for handling continuous phase shift design in this section can be categorized into relaxation methods (SDR and CR), iterative approximation methods (the penalty-based method and MM), and gradient methods (GD and the manifold method). Their relationships are summarized in Figure 6, and their properties are compared in Table 3.  
To compare the performance of different optimization methods, the three application examples mentioned in Section 2 are simulated under phase shift model C1. All simulations are performed on MATLAB R2017a on a Windows X64 desktop with 3.2 GHz CPU and 16 GB RAM. For fair comparisons, all algorithms start from the same initial point (any  
Figure 5: Graphical illustration of the Riemannian CG method at the \(l^{th}\) iteration. A similar figure can be found in the optimization literature, e.g., [47].  
feasible point can serve as an initial point), and the stopping criterion for iterative methods is when the relative change of two consecutive objective function values becomes less than \(10^{-4}\), and the maximum number of iterations for all methods is set to 100. By employing the BCD framework for solving for \(\mathbf{x}\) and \(\mathbf{e}\) in (4), the three applications can be efficiently solved, and the simulation results are shown in Figure 7a-c, respectively.  
Figure 6: Relationships among different optimization methods.  
Figure 7: Performance comparisons of six optimization methods with \(M=10\): (**a**) Secrecy rate versus the maximum transmit power under the number of BS antennas \(N=20\)[20]. (**b**) Uplink transmit power versus the number of users under the number of BS antennas \(N=20\) and transmission power limitation \(P_{k}=10.8\) dBm [21]. (**c**) Total network cost versus the number of users under the number of BS antennas \(N=10\), the target rate \(R_{k}=10\) MHz, the bandwidth \(B=10\) MHz, the regularization parameter \(\eta=100\) and the local storage size \(S_{\max}=100\)[22].  
From these figures, it can be observed that, out of the six algorithms, GD and the manifold method perform consistently well in all three applications, followed by the MM method and the penalty method. On the other hand, the SDR method and CR perform the worst in these three applications. The worse performance of the SDR method and CR is due to the relatively weak guarantee in the solution quality.  
On the other hand, the computation times of various methods in the first application are shown in Figure 8a. From this figure, it can be seen that the manifold method, the GD method, and the CR-PG method require the least amounts of computation time among the six algorithms, achieving at least two orders of magnitude reduction compared with the SDR method and the penalty method when \(N>50\). This advantage becomes more prominent as the number of reflecting elements \(M\) increases as shown in Figure 8b. The computation times for the other two applications show similar behaviors and thus are not shown here.