# Energy-Efficient RIS-Enabled SISO-OFDMA Communication via Lower Bound Optimization
## 3 Proposal Method and Discussion
### Optimization with Respect to RIS Phase Shift Matrix  
Consider the optimized values for the vectors \(\mathbf{\rho}\) and \(\mathbf{P}\) obtained in the first step of the AO algorithm. The EE optimization problem with respect to RIS phase shifts is formulated as follows:  
\[\underset{\mathbf{\Phi}}{\text{maximize}} \text{EE}=\frac{\sum_{k=1}^{K}\sum_{n=1}^{N}\rho_{k,n}r_{k,n}}{ \xi\sum_{k=1}^{K}p_{k}+P_{\text{BS}}+KP_{\text{UE}}+MP_{m}(b)}\] (16) subject to \[|\phi_{m}|=1,\quad\forall n=1,2,\ldots,M. \tag{16a}\]  
In this problem, \(EE\) represents Energy Efficiency, \(\mathbf{\Phi}\) is the RIS phase shift matrix, and the constraints in (16a) ensure that the phase shifts have unit magnitude. This EE maximization problem can be transformed into a rate maximization problem with respect to the RIS phase shift variables. \[\underset{\mathbf{\Phi}}{\text{maximize}} R_{\text{EE}}=\sum_{k=1}^{K}\sum_{n=1}^{N}\rho_{k,n}r_{k,n}\] (17a) subject to \[|\phi_{m}|=1,\quad\forall n=1,2,\ldots,M. \tag{17b}\]To tackle this optimization problem, we employ the PSO algorithm, renowned for its versatility in solving complex optimization tasks. PSO maintains a population of particles, each associated with a phase shift vector, and efficiently explores the solution space to find optimal solutions. Here is a more detailed breakdown of the key operations and parameters in the PSO algorithm. ```
1. Choose parameters \(\eta\in(0,0.5)\), \(\tau\in(0,1)\), and an initial pair \((x^{(0)},\lambda^{(0)})\);
2. Evaluate \(g^{(0)},\
abla g^{(0)},b^{(0)},A^{(0)}\);
3. If a quasi-Newton approximation is used, choose an initial \(n\times n\) symmetric positive definite Hessian approximation \(\mathbf{B}^{(0)}\), otherwise compute \(\
abla^{2}_{xx}\mathcal{L}^{(0)}\);
4.repeat until a convergence test is satisfied
5. Compute \(\mathbf{d}^{(t)}\) by solving (14); let \(\hat{\lambda}\) be the corresponding multiplier;
6. Set \(\mathbf{d}_{\lambda}\leftarrow\hat{\lambda}-\lambda^{(t)}\);
7. Choose \(\mu^{(t)}\) to satisfy \(\mu\geq\frac{\
abla f_{i}^{\prime}p_{i}+\xi p_{i}^{\prime}\
abla^{2}_{xx}l_{i} p_{i}}{(1-\rho)\|\alpha_{i}\|_{1}}\) with \(\sigma=1\);
8. Set \(\mathbf{\alpha}^{(t)}\gets 1\);
9. while \(O(x^{(t)}+\mathbf{\alpha}^{(t)}d^{(t)};\mu^{(t)})\leq O(x^{(t)};\mu^{(t)})+\eta_{ \mathbf{\alpha}^{(t)}}D(O(x^{(t)};\mu^{(t)}),d^{(t)})\)
10. Reset \(\mathbf{\alpha}^{(t)}\leftarrow\tau^{(t)}\mathbf{\alpha}^{(t)}\) and \(\lambda^{(t+1)}\leftarrow\lambda^{(t)}+\mathbf{\alpha}^{(t)}d_{\lambda}\);
11. endwhile
12. \(\mathbf{x}^{(t+1)}\leftarrow\mathbf{x}^{(t)}\mathbf{\alpha}^{(t)}d^{(t)}\) and \(\lambda^{(t+1)}\leftarrow\lambda^{(t)}+\mathbf{\alpha}^{(t)}\mathbf{d}_{\lambda}\);
13. Evaluate \(g^{(t+1)},\
abla g^{(t+1)},b^{(t+1)},\mathbf{A}^{(t+1)}\), (and possibly \(\
abla^{2}_{xx}\mathcal{L}^{(t+1)}\));
14. If a quasi-Newton approximation is used, set
15. \(\mathbf{d}^{(t)}\leftarrow\mathbf{\alpha}^{(t)}d^{(t)}\) and \(\mathbf{y}^{(t)}\leftarrow\
abla_{x}\mathcal{L}(x^{(t+1)},\lambda^{(t+1)})-\
abla^ {x}\mathcal{L}(x^{(t)},\lambda^{(t+1)})\),
16. and obtain \(\mathbf{B}^{(t+1)}\) by updating \(\mathbf{B}^{(t)}\) using a quasi-Newton formula;
17.repeat
```  
**Algorithm 2**Sequential fractional programming algorithm. The PSO algorithm, outlined in Algorithm 3, efficiently explores the solution space by maintaining a population of particles with associated phase shift vectors. The algorithm's complexity is proportional to the product of the population size \(L\) and the maximum number of iterations \(T\). For each particle \(i=1,2,\ldots,L\) at time \(t\), it is associated with a \(1\times M\) phase shift vector \(\mathbf{\theta}^{(t)}_{i}=(\theta_{i1},\theta_{i2},\ldots,\theta_{iM})\), where each component is limited within \([-\
u_{\max},\
u_{\max}]\).