## 4 Solution of the Problem
### UAV Deployment Optimization
For variable \(\left\|v_{k}^{H}\mathbf{\Phi}_{k}\right\|^{2}\), problem (17a) is a monotonically increasing function. The proof of monotonicity is given in Appendix A. Therefore, problem (17a) can be expressed as follows  
\[\max_{\boldsymbol{v}_{k}}\sum_{k\in\{t,r\}}\boldsymbol{v}_{k}^{H} \mathbf{\Phi}_{k}\mathbf{\Phi}_{k}^{H}\boldsymbol{v}_{k}\] (18a) s.t. \[\text{ (\ref{eq:16a}),(\ref{eq:16a}), (\ref{eq:16a}), (\ref{eq:16a}), (\ref{eq:16a}),} \tag{18b}\]  
where problem (18a) is a non-convex quadratically constrained quadratic program (QCQP), and can be reformulated as a homogeneous QCQP as follows:  
\[\max_{\overline{\boldsymbol{v}}_{k}}\sum_{k\in\{t,r\}} \overline{\boldsymbol{v}}_{k}^{H}\mathcal{H}\overline{\boldsymbol{v}}_{k}\] (19a) s.t. \[\text{ (\ref{eq:16a}),(\ref{eq:16a}), (\ref{eq:16a}), (\ref{eq:16a}),} \tag{19b}\]  
where \(\overline{\boldsymbol{v}}_{k}=\boldsymbol{v}_{k}\), \(\mathcal{H}=\mathbf{\Phi}_{k}\mathbf{\Phi}_{k}^{H}\), and \(\overline{\boldsymbol{v}}_{k}^{H}\mathcal{H}\overline{\boldsymbol{v}}_{k}= \text{Tr}\!\left(\mathcal{H}\overline{\boldsymbol{v}}_{k}\overline{\boldsymbol{v }}_{k}^{H}\right)\). Problem (19a) is a non-deterministic polynomial that is hard in general [20]. We assume \(\mathbf{V}_{k}=\overline{\boldsymbol{v}}_{k}\overline{\boldsymbol{v}}_{k}^{H}\), which needs to satisfy \(\text{Rank}(\mathbf{V}_{k})=1\) and \(\mathbf{V}_{k}\succeq 0\). Then, we apply SDR to relax the rank-one constraint \(\mathbf{V}_{k}\). As a result, problem (19a) is simplified to  
\[\max_{\mathbf{V}_{k}}\sum_{k\in\{t,r\}}\text{Tr}\!\left(\mathcal{H }\mathbf{V}_{k}\right)\] (20a) s.t. \[\text{Rank}(\mathbf{V}_{k})=1,\mathbf{V}_{k}\succeq 0,\text{(\ref{eq:16a}), (\ref{eq:16a}), (\ref{eq:16a}), (\ref{eq:16a}).} \tag{20b}\]  
It is clear that problem (20a) is a standard convex semidefinite program that can be solved by convex optimization solvers such as CVX. However, the solution of the relaxed problem may not be a rank-one solution, i.e., \(\text{Rank}(\mathbf{V}_{k})\ \
eq 1\), which indicates that the solution of (20a) only stands for an upper bound of problem (19a). In order to construct a rank-one solution from the optimal higher-rank solution to problem (20a). We also need a couple of additional steps. First, we obtain the eigenvalue decomposition of \(\mathbf{V}_{k}\) as \(\mathbf{V}_{k}=\mathbf{U}_{k}\mathbf{\Sigma}_{k}\mathbf{U}_{k}^{H}\), where \(\mathbf{U}_{k}=\left[e_{1}^{k},\cdots,e_{M}^{k}\right]\) and \(\mathbf{\Sigma}_{k}=\text{diag}\!\left(\lambda_{1}^{k},\cdots,\lambda_{M}^{k}\right)\) are a unitary matrix and diagonal matrix, respectively. Second, we obtain a suboptimal solution to (19a) as \(\overline{\boldsymbol{v}}_{k}=\mathbf{U}_{k}\!\times\!\!\mathbf{\Sigma}_{k} \mathbf{\tau}_{k}\), where \(\mathbf{\tau}_{k}\in\mathbb{C}^{M\times 1}\) is a random vector generated according to \(\mathbf{r}_{k}\sim\mathcal{CN}(0,\mathbf{I}_{M})\) with \(\mathcal{CN}(0,\mathbf{I}_{M})\) denoting the the circularly symmetric complex Gaussian distribution with zero mean and covariance matrix \(\mathbf{I}_{M}\). With independently generated Gaussian random vectors \(\mathbf{r}_{k}\), the objective value of (19a) is approximated as the maximum one attained by the best \(\overline{v}_{k}\) among all \(\mathbf{r}_{k}\) values. Finally, the solution \(\overline{v}_{k}\) to problem (19a) can be recovered by \(\overline{v}_{k}=\ell^{\text{arg}([\frac{\overline{v}_{k}}{M_{k}}]_{(1:M)})}\), where \([\mathbf{z}]_{(1:M)}\) denotes the vector that contains the first \(M\) elements in \(\mathbf{z}\). It has been shown that such an SDR approach followed by a sufficiently large number of randomizations of \(\mathbf{r}_{k}\) guarantees a \(\overline{\overline{\alpha}}\)-approximation of the optimal objective value of problem (19a) [16]. The following algorithm is proposed to implement the above solution. First, we initialize \(\theta_{uav}\), \(h_{uav}\), \(U_{density}\), \(\mathbf{w}_{k}\), \(\mathbf{\Theta}_{k}\), and \(h_{uav}\).