## 4 Joint Optimization of Altitude and Energy Beamforming for Multiple EHDs
### Efficient Near-Optimal Altitude and Energy Beamforming Design with Low Complexity  
In this subsection, we propose a low-complexity method of obtaining a near-optimal solution by avoiding the 1D exhaustive line search method, which results in high complexity when using an SDP solver. Motivated by insights from the method of obtaining the near-optimal solution to the single-EHD case in Section 3, we devise the golden-section (GS) line search method with search space reduction, which determines the solution much faster. The details are as follows. At the \(u\)-th EHD, from Lemma 1, the optimal energy beamforming to minimize the expected required charging time, denoted by \(\mathbf{X}_{u}^{\star}\), is given by  
\[\mathbf{X}_{u}^{\star}=P\mathbf{\mathbbm{a}}_{u}\mathbf{\mathbbm{a}}_{u}^{H}, \ \forall u\in\mathcal{U}. \tag{46}\]  
Moreover, the near-optimal UAV altitude to minimize the required charging time, denoted by \(h_{\text{Prop},\mu}^{\star}\), is obtained from Algorithm 1. When a UAV adopts an altitude and energy beamforming as the optimal altitude and energy beamforming, respectively, for the \(u\)-th EHD, i.e., \(h=h_{\text{Prop},\mu}^{\star}\) and \(\mathbf{X}=\mathbf{X}_{u}^{\star}\) at the UAV, the expected required charging time at the \(k\)-th EHD (\(k\in\mathcal{U}\)), denoted by \(\tau_{k}^{\star,\mu}(h_{\text{Prop},\mu}^{\star},\mathbf{X}_{u}^{\star})\), then becomes  
\[\tau_{k}^{\star,\mu}(h_{\text{Prop},\mu}^{\star},\mathbf{X}_{u}^{\star})=\frac {Q_{k}^{\text{Req}}}{\text{Tr}\left(\mathbf{G}_{k}(h_{\text{Prop},\mu}^{\star })\mathbf{X}_{u}^{\star}\right)},\ \forall k\in\mathcal{U}. \tag{47}\]  
In this case, the overall charging time required to meet the energy demands of all EHDs (i.e., \(\forall k\in\mathcal{U}\)), denoted by \(t_{\star,\mu}^{\max}\), is expressed as  
\[t_{\star,\mu}^{\max}=\max\left(\tau_{1}^{\star,\mu}(h_{\text{Prop},\mu}^{\star},\mathbf{X}_{u}^{\star}),...,\tau_{U}^{\star,\mu}(h_{\text{Prop},\mu}^{\star},\mathbf{X}_{u}^{\star})\right),\ \forall u\in\mathcal{U}, \tag{48}\]  
which signifies the overall charging time corresponding to the pair \((h_{\text{Prop},\mu}^{\star},\mathbf{X}_{u}^{\star})\) at the UAV. Among all the overall charging times obtained from (48), i.e., \(t_{\star,1}^{\max}\), \(t_{\star,2}^{\max}\),..., and \(t_{\star,U}^{\max}\), we can identify the lowest time among those achieved by all pairs, i.e., \((h_{\text{Prop},1}^{\star},\mathbf{X}_{1}^{\star})\), \((h_{\text{Prop},2}^{\star},\mathbf{X}_{2}^{\star})\),..., and \((h_{\text{Prop},1}^{\star},\mathbf{X}_{U}^{\star})\), as follows:  
\[u_{\min,1}=\arg\min_{u\in\mathcal{U}}\bigl{(}t_{\star,1}^{\max}\,,...,t_{ \star,\mu}^{\max}\,,...,t_{\star,U}^{\max}\bigr{)}, \tag{49}\]  
with its corresponding pair \((h_{\text{Prop},u_{\min,1}}^{\star},\mathbf{X}_{\min,1}^{\star})\). From (49), we infer that the overall charging time is most effectively reduced for \(h_{\text{Prop},u_{\min,1}}^{\star}\) compared with the other pairs \([h_{\text{Prop},k}^{\star}]_{k\
eq u_{\min,1}}^{\text{II}}\). Moreover, the second lowest time is also obtained using  
\[u_{\min,2}=\arg\min_{u\in\mathcal{U}}\bigl{(}t_{\star,1}^{\max}\,,...,t_{ \star,\mu}^{\max}\,,...,t_{\star,U}^{\max}\bigr{)}, \tag{50}\]  
with its corresponding pair \((h_{\text{Prop},u_{\min,2}}^{\star},\mathbf{X}_{u_{\min,2}}^{\star})\). From (49) and (50), we determine the effective altitude range between \(h_{\text{Prop},u_{\min,1}}^{\star}\) and \(h_{\text{Prop},u_{\min,2}}^{\star}\), which is represented by  
\[\Big{[}\min\,\bigl{(}h_{\text{Prop},u_{\min,1}}^{\star},h_{\text{Prop},u_{\min, 2}}^{\star}\bigr{)},\max\bigl{(}h_{\text{Prop},u_{\min,1}}^{\star},h_{\text{Prop}, u_{\min,2}}^{\star}\bigr{)}\Big{]}. \tag{51}\]  
The overall charging time cannot be effectively reduced for altitudes outside this range, which is taken into account in our proposed method, where the search space \([H_{\min},H_{\max}]\) is reduced.