## 3 Optimization Problem Solving
### Convergence and Algorithm Complexity Analysis  
The proposed alternating optimization-based energy efficiency (AO-EE) optimization algorithm is shown in Algorithm 3. ```
1:Initialization:\(\mathbf{f}_{0}\), \(\mathbf{V}_{0}\) is randomly generated, \(i=0\), \(\varepsilon=10^{-3}\);
2:repeat
3: Calculate \(\mathbf{V}_{i+1}\) based on Algorithm 1 with fixed \(\mathbf{f}_{i}\);
4: Calculate \(\mathbf{f}_{i+1}\) based on Algorithm 1 with fixed \(\mathbf{V}_{i}\);
5:Until The increase of the objective value of the problem P1 is below the threshold \(\varepsilon\). ```  
**Algorithm 3** AO-EE Algorithm for problem P1. We define \(\mathbf{f}_{i}\), \(\mathbf{V}_{i}\) as the i-th iteration solution of the problems (P2), (P3). The objective function is denoted by \(EE(\mathbf{f}_{i},\mathbf{V}_{i})\). In step 3 of Algorithm 3, the BS transmit beamforming vector can be obtained for a given \(\mathbf{V}_{i}\). Hence, we have:  
\[EE(\mathbf{f}_{i},\mathbf{V}_{i})\leq EE\big{(}\mathbf{f}_{i+1},\mathbf{V}_{i}\big{)} \tag{32}\]  
In step 4 of Algorithm 3, we can obtain an IRS phase shift when \(\mathbf{f}_{i}\) is given. We have:  
\[EE\big{(}\mathbf{f}_{i+1},\mathbf{V}_{i}\big{)}\leq EE\big{(}\mathbf{f}_{i+1},\mathbf{V}_{i+1} \big{)} \tag{33}\]  
Thus, we obtain:  
\[EE(\mathbf{f}_{i},\mathbf{V}_{i})\leq EE\big{(}\mathbf{f}_{i+1},\mathbf{V}_{i+1}\big{)} \tag{34}\]  
Therefore, \(EE(\mathbf{f}_{i},\mathbf{V}_{i})\) is non-decreasing over the iterations in the AO-EE, and \(EE(\mathbf{f}_{i},\mathbf{V}_{i})\) has the upper bound due to the transmit power constraint. Thus, the AO-EE algorithm is guaranteed to converge. To validate the effectiveness of the proposed AO-EE algorithm, we compare it with five benchmark algorithms, as follows:  
1. No-IRS scheme: this scheme does not consider IRS-related channels and only optimizes the BS transmit beamforming using the Dinkelbach-SCA algorithm, denoted as \"NO-IRS\". Random phase-shift scheme: in this scheme, the phase of each reflecting element is uniformly and independently generated from [0, 2\(\pi\)] randomly, and the BS transmit beamforming is optimized using the Dinkelbach-SCA algorithm, denoted as the \"random phase\". 3. SDR scheme: the IRS phase-shift optimization is conducted using the SDR algorithm [28], while the BS transmit beamforming is optimized using the Dinkelbach-SCA algorithm, denoted as \"baseline scheme1\". 4. Phase alignment scheme: the IRS phase-shift optimization is performed using phase alignment [28], and the BS transmit beamforming is optimized using the Dinkelbach-SCA algorithm, denoted as \"Baseline scheme2\". 5. SDR-SDR scheme: Both the BS transmit beamforming and IRS phase shift are optimized using the SDR algorithm [37], denoted as \"Baseline scheme3\". The complexity analysis of the AO-EE algorithm is as follows: to solve the BS transmit beamforming, the Dinkelbach and SCA algorithms are used, and the complexity of solving problem P2-3 is \(\mathcal{O}\big{(}M^{3}\big{)}\)[38]. The IRS phase shift is optimized using the Riemannian Conjugate Gradient (RCG) algorithm, where the main complexity lies in calculating the Euclidean gradient, which is \(\mathcal{O}\big{(}N^{2}\big{)}\). The backtracking step also requires an iterative search for \(\tau_{2}\), but its complexity can be neglected when N is large. Therefore, the complexity of the RCG algorithm is \(\mathcal{O}\big{(}N^{2}\big{)}\)[39]. Let the iterations of the Dinkelbach algorithm, SCA algorithm, and alternating optimization be \(L_{1}\), \(L_{2}\), and \(I_{iter}\), respectively. The overall complexity of the AO-EE algorithm is \(\mathcal{O}\Big{(}I_{iter}\big{(}L_{1}L_{2}M^{3}+N^{2}\big{)}\log_{2}\Big{(} \frac{1}{\varepsilon}\Big{)}\Big{)}\), where \(\varepsilon\) is the convergence threshold.