# Phase Shift Design in RIS Empowered Wireless Networks: From Optimization to AI-Based Methods
## 3 Review on Optimization Methods under Continuous Phase Shift
### Penalty Method  
To guarantee a feasible solution while avoiding the feasibility check of the SDR method, a penalty method can be employed. To be specific, the rank constraint \(\text{rank}(\mathbf{Q})=1\) in (6)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimization Methods** & **Property of Solutions** & **Complexity Order** & **Applicable Model** & **Examples** \\ \hline SDR & infeasible\(/\)feasible solution & \(\mathcal{O}\Big{(}\sqrt{M}(2M^{4}+M^{3})\Big{)}\) & C1 and C2 & [23, 24] \\ \hline Penalty & stationary solution & \(\mathcal{O}(M^{3})\) & C1 and C2 & [25, 26] \\ \hline MM & locally optimal solution [13] & \(\mathcal{O}\big{(}M^{2}\big{)}\) & C1 and C2 & [3, 27] \\ \hline GD & stationary solution [15] & \(\mathcal{O}(M)\) & C1, C2, and C3 & [28, 29] \\ \hline Manifold & stationary solution [14] & \(\mathcal{O}(M)\) & C1 and C2 & [22, 30] \\ \hline CR & feasible solution [15] & \(\begin{array}{c}\mathcal{O}\big{(}M^{3}\big{)}\text{ using CVX}\\ \mathcal{O}(M)\text{ using PG}\end{array}\) & C1 and C2 & [16, 31] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of optimization methods for continuous phase shift RIS.  
can be equivalently expressed as \(\text{Tr}(\sqrt{\mathbf{Q}^{*}\mathbf{Q}})-\|\mathbf{Q}\|_{2}\leq 0\)[46], where \(\mathbf{Q}^{*}\) is the conjugate of \(\mathbf{Q}\). Then, with the constraint added as a penalized term, this further transforms problem (6) into  
\[\underset{\mathbf{Q}:\mathbf{Q}=\mathbf{0}}{\text{min}} f\Big{(}\mathbf{x}^{(n)},\mathbf{Q}\Big{)}+\frac{1}{\mu}\Big{(} \text{Tr}(\sqrt{\mathbf{Q}^{*}\mathbf{Q}})-\|\mathbf{Q}\|_{2}\Big{)},\] s.t. \[\mathbf{Q}_{m,m}=1,\ \ \forall m, \tag{7}\]  
where \(\mu\in(0,1)\) is a penalty factor penalizing the violation of constraint \(\text{Tr}(\sqrt{\mathbf{Q}^{*}\mathbf{Q}})-\|\mathbf{Q}\|_{2}\leq 0\). This transformed objective function now contains a difference-of-convex (DC) term \(\text{Tr}(\sqrt{\mathbf{Q}^{*}\mathbf{Q}})-\|\mathbf{Q}\|_{2}\). To convert the DC term to a convex form, SCA can be applied to \(-\|\mathbf{Q}\|_{2}\) (if \(f\) is non-convex, the SCA can also be applied to \(f\) at the same time).  
The resulting problem is convex in \(\mathbf{Q}\) if \(f\Big{(}\mathbf{x}^{(n)},\mathbf{Q}\Big{)}\) is convex. Accordingly, the optimal \(\mathbf{Q}\) in each SCA iteration can be obtained by employing the interior-point method. Since the transformed problem is solved under the SCA framework, a stationary solution of \(\mathbf{Q}\) can be guaranteed. Furthermore, since problem (5) is equivalent to the transformed problem as \(\mu\) tends to zero, the obtained solution is also a stationary point to (5). The penalty factor \(\mu\) is important in controlling how strict the rank constraint is imposed. In practice, it can be a decreasing sequence with respect to the SCA iteration to guarantee a feasible solution of (5) at the end of the iteration. As the interior-point method is adopted in each SCA iteration, the complexity order is at least \(\mathcal{O}(M^{3})\).