# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 3 Joint Optimization Algorithm
### Transmit Beamforming Optimization  
Given the UAV position \(\mathbf{q}\), the optimization subproblem for the beamforming \(\omega_{k}^{c},\forall k\) and \(\mathbf{\omega}_{n}^{r},\forall n\) can be formulated as:  
\[ \begin{split}\text{P2}:\min_{\omega_{k}^{c},\omega_{n}^{c}}\max _{j,k}\Bigl{(}\text{SINR}_{j,k}^{c}\Bigr{)}\\ \text{s.t.}\ \ \text{C1},\text{C2},\text{C3}\end{split}  \tag{12}\]Notice that problem P2 is an MM (minorize maximization) problem. To simplify the solution, we introduce an auxiliary variable \(A\), transforming the MM problem into a minimization problem. The reformulated problem P2 is expressed as:  
\[ \begin{split}&\text{P3}:\min_{\omega_{k}^{r},\omega_{k}^{r},A}A\\ &\text{s.t.}&\text{C4}:\text{SINR}_{j,k}^{r}=\left| \begin{subarray}{c}g_{i}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{k}^{r}\\ \sum\limits_{i=1,j+k}^{K}g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{i}^{ r}\\ &+\sum\limits_{n=1}^{N}g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{ \omega}_{n}^{r}\\ &\text{C1},\text{C2},\text{C3}\end{subarray}\right.\leq A,\forall j,k \\ \end{split}  \tag{13}\]  
This is a minimization problem, but due to the single fractional constraint C4 and non-convex constraints C1 and C2, problem P3 remains difficult to solve. Noticing that problem P3 has a similar fractional programming form to those in [10; 17], we can apply the Dinkelbach transformation to convert it into a solvable form, i.e., transforming the fractional constraint into a polynomial expression. By introducing an auxiliary variable \(c_{j,k}\), constraint C4 is rewritten as:  
\[\left|g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{k}^{r}\right|^{2}-c_{j,k} \left(\left|\sum\limits_{i=1,j+k}^{K}g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{ \omega}_{i}^{r}\right|^{2}+\left|\sum\limits_{n=1}^{N}g_{j}^{\text{H}}(\mathbf{q}, \mathbf{e}_{j})\mathbf{\omega}_{n}^{r}\right|^{2}+\sigma_{e}^{2}\right)\leq A,\forall j,k \tag{14}\]  
The auxiliary variable \(c_{j,k}\) essentially represents the eavesdropping SINR of the eavesdropper \(j\) for the user \(k\)[17], and it can be updated using the transmit beamforming \(\mathbf{\omega}_{k}^{r},\forall k\) and \(\mathbf{\omega}_{n}^{r},\forall n\). When the transmit beamforming is given, the optimal \(c_{j,k}^{*}\) can be expressed as:  
\[c_{j,k}^{*}=\frac{\left|g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{k}^{r} \right|^{2}}{\left|\sum\limits_{i=1,j+k}^{K}g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j })\mathbf{\omega}_{i}^{r}\right|^{2}+\left|\sum\limits_{n=1}^{N}g_{j}^{\text{H}}( \mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{n}^{r}\right|^{2}+\sigma_{e}^{2}},\forall j,k \tag{15}\]  
It can be seen that constraints C1, C2, and C3 contain quadratic variables \(\mathbf{\omega}_{k}^{r},\forall k\) and \(\mathbf{\omega}_{n}^{r},\forall n\), making the problem difficult to handle. Therefore, we adopt the SDR method. We define \(\mathbf{W}_{k}^{\ast}\triangleq\mathbf{\omega}_{k}^{r}\mathbf{\omega}_{k}^{r}\)H, \(\forall k\) and \(\mathbf{W}_{n}^{\ast}\triangleq\mathbf{\omega}_{n}^{r}\mathbf{\omega}_{n}^{r}\)H, \(\forall n\), with the rank-one constraint expressed as:  
\[ \begin{split}&\text{Rank}(\mathbf{W}_{k}^{\ast})=1,\forall k\\ &\text{Rank}(\mathbf{W}_{n}^{\ast})=1,\forall n\end{split}  \tag{16}\]  
Problem P3 can be transformed into:  
\[\begin{split}&\text{P4}:\min_{\mathbf{W}_{k}^{r},\mathbf{W}_{n}^{r}A}A\\ &\text{s.t.