# Energy-Efficient RIS-Enabled SISO-OFDMA Communication via Lower Bound Optimization
## 3 Proposal Method and Discussion
### Optimization Respect to Subcarrier and Power Allocations
#### 3.1.1 Subcarrier Allocations  
Utilizing a heuristic algorithm is advantageous in this scenario since the maximum achievable energy efficiency of the \(k_{\text{th}}\) UE, \(\hat{\text{EE}}=\frac{\sum_{k\in\mathcal{N}}\rho_{k,n}r_{k,n}}{\sum_{k\in \mathcal{N}}\zeta p_{k,n}+\alpha_{k,\text{RIS}}\hat{R}_{k}}\), depends solely on its own parameters and the subcarriers it occupies, without being influenced by the power adaptation strategies of other users. To simplify, we assume there is no difference in the RIS configuration matrix between subcarriers of the same user, meaning \(\mathbf{\Phi}_{1}^{k}=,\cdots=\mathbf{\Phi}_{m}^{k}=\mathbf{\Phi}^{k}\). Additionally, constant values for \(\mathbf{\alpha}_{k,\text{RIS}}\) are considered for further simplification. Negative values of \(\mathbf{\alpha}_{k,\text{RIS}}\) may arise due to the nature of \(\alpha_{k,\text{RIS}}P_{c}\), representing the hypothetical static circuit power allocated to the \(k_{\text{th}}\) UE and \(n_{\text{th}}\) RIS element. It is important to note that this value serves as a conceptual representation rather than a concrete physical reality. In practice, the total static circuit power remains constant at \(P_{c}\). To solve the optimization problem in Equation (10), we employ a heuristic algorithm called MDSA [22]. The power allocation is optimized using the SQP method, which is integrated into the MDSA algorithm. The central concept behind the MDSA algorithm is to iteratively assign subcarriers, with the aim of maximizing the minimum individual EE, denoted as EE, while ensuring that QoS requirements are met. Initially, each UE virtually receives its worst subcarrier, a conservative starting point. In this scenario, an individual EE is optimized under the QoS constraint using the algorithm employed for the power allocation, as described in Lines 2 to 5. In each iteration, the UE with the lowest individual EE selects its preferred subcarrier from the pool of unassigned subcarriers. The UE then optimizes its individual EE under the QoS requirements using the SQP method. This iterative process continues until all subcarriers have been assigned, as outlined in lines 7 to 12 of Algorithm 1. ```
1.Input:\(\boldsymbol{\rho}=[\rho_{k,n}]_{K\times N}\gets 0_{K\times N}\); \(\varrho_{k}\leftarrow\emptyset\), \(\forall k\in\mathcal{K}\); \(\boldsymbol{\alpha}_{k,\text{RIS}}\leftarrow\alpha_{k,\text{RIS}}^{\text{ini}}\)  2.Output:\(\boldsymbol{\rho},\boldsymbol{P}\)  3.For each UE \(k\in\mathcal{K}\)  4. Find the subcarrier \(\hat{n}_{k}\leftarrow\arg\min_{n\in\mathcal{N}}\text{SNR}_{k,n}\)  5. Calculate \(\text{EE}\leftarrow\max_{\hat{n}_{k}\geq R_{k}^{-1}}\frac{R_{k}(\hat{n}_{k}, \hat{n}_{k})}{\hat{n}_{k}+\hat{n}_{k}\text{ES}\hat{P}_{c}}\)  6.End  7.While\(\mathcal{N}\
eq\phi\)  8. Find the UE \(k\in\mathcal{K}\) such that \(\hat{k}\leftarrow\arg\min_{k\in\mathcal{K}}\text{EE}\)  9. Find the UE \(\hat{n}\in\mathcal{N}\) such that \(\hat{n}\leftarrow\arg\max_{n\in\mathcal{N}}\text{SNR}_{k,n}\)  10. Set \(\rho_{k,n}\gets 1\); \(\varrho_{k}\gets\varrho_{\hat{k}}\cup\hat{n}_{\hat{k}}\); \(\mathcal{N}\leftarrow\mathcal{N}\backslash\{\hat{n}_{\hat{k}}\}\)  11. Calculate \(\text{EE}\leftarrow\max_{\hat{P}_{k}\geq R_{k}^{-1}}(\varrho_{k},R_{k})\frac{R_{ k}(\hat{n}_{k},\hat{P}_{k})}{\hat{n}_{k}+\hat{n}_{k}\text{ES}\hat{P}_{c}}\)  12.End
```  
**Algorithm 1**MDSA Maximizing-EE-lower-bound-based downlink subcarrier assignment. Notably, the MDSA algorithm does not impose a total transmit power constraint explicitly. However, this constraint is likely to be automatically satisfied during subcarrier allocation for several reasons:  
By appropriately choosing the parameter \(\alpha_{k,\text{RIS}}\), the eventually optimized individual EEs tend to converge to similar values. If a UE initially requires excessive power to meet its QoS, it likely has a lower individual EE and will request more subcarriers later. This allows the UE to reduce its transmit power and increase its EE.