# Energy-Efficient RIS-Enabled SISO-OFDMA Communication via Lower Bound Optimization
## 3 Proposal Method and Discussion
### Optimization Respect to Subcarrier and Power Allocations  
It is difficult to optimize the original objective function in (8) directly, but it may be possible to optimize a surrogate objective function that is easier to handle. In this case, the lower bound of the objective function can be used as a surrogate objective function to facilitate the optimization process. The surrogate objective function is then optimized instead of the original objective function, and the solution obtained using the surrogate objective function is used as a lower bound on the optimal value of the objective function.  
Based on the insights from the work conducted in [22], the following theorem and corollary are established.  
**Theorem 1**.: _By considering a fixed value for the RIS phase shifts and introducing a new parameter \(\mathbf{\kappa}_{k,\text{RIS}}=\mathbf{\alpha}_{k,\text{LIE}}\times\mathbf{\alpha}_{m, \text{RIS}}\) into Equation (8), the optimal EE is given by:_  
\[(P2) \quad\hat{E}\hat{E}=\max_{\begin{subarray}{c}\rho\in\mathcal{G}, \mathbf{P}\in\mathcal{P},\\ \mathbf{\kappa}\in\mathcal{G},\end{subarray}}\min_{k\in\mathcal{K}}\frac{\sum_{k \in\mathcal{K}}\sum_{n\in\mathcal{N}}\rho_{k,n}r_{k,n}}{\sum_{k\in\mathcal{K} }\sum_{n\in\mathcal{N}}\zeta p_{k,n}+\alpha_{k,\text{RIS}}\times P_{c}}\] (9) \[\quad s.t.\sum_{n\in\mathcal{N}}\rho_{k,n}r_{k,n}\geqslant\hat{R} _{k},\quad\forall k=1,2,\ldots,K.\] (9a) _where \[\mathbf{\alpha}_{k,\text{RIS}}=\{[\mathbf{\alpha}_{k,\text{LIE}}]_{\text{K}\times 1 },[\mathbf{\alpha}_{m,\text{RIS}}]_{\text{M}\times 1}\mid\sum_{k\in\mathcal{K}} \sum_{n\in\mathcal{N}}\alpha_{k,\text{RIS}}=1;\mathbf{\alpha}_{k,\text{RIS}}\in \mathbb{R}\}\] and \[P_{c}=P_{BS}+KP_{\text{LIE}}+MP_{m}(b)\] _. The structure of the optimal solution, as demonstrated in Theorem_ 1 _presented in Appendix A, can be depicted in a split form. This split form provides valuable insights that enable the derivation of Corollary_ 1_._  
**Corollary 1**.: _For any fixed \(\mathbf{\alpha}_{k,\text{RIS}}\in\mathbf{\alpha}_{k,\text{RIS}}\), the optimal energy efficiency (EE) in (9) is bounded from below by:_  
\[(P3) \quad\hat{E}\hat{E}\geq\max_{\rho\in\mathcal{G},\mathbf{P}\in\mathcal{ P}}\min_{k\in\mathcal{K}} \left\{\frac{\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}\rho_{k,n}r_{k,n}}{ \sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}\zeta p_{k,n}+\alpha_{k,\text{RIS} }\times P_{c}}\right\}\] (10) _subject to_ \[\quad\sum_{n\in\mathcal{N}}\rho_{k,n}r_{k,n}\geq\hat{R}_{k},\forall k =1,2,...,K. \tag{10a}\]  
Assuming that \(\mathbf{\alpha}_{k,\text{RIS}}\text{opt}=[\mathbf{\alpha}_{k,\text{RIS}}^{\text{opt}}]_ {\text{K}\times 1}\) corresponds to the optimal energy efficiency (EE) in Equations (8) and (9), \(\mathbf{\alpha}_{k,\text{RIS}}^{\text{opt}}\) can be interpreted as the portion of static circuit power individually allocated to the \(k_{\text{th}}\) UE and the \(m_{\text{th}}\) RIS element when aiming for maximum EE.  
This decomposition allows us to split the complex optimization objective in Equation (8) into a set of relatively independent and simpler objectives in Equation (10). Consequently, we can employ sub-optimal methods for efficient problem-solving.