# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 3 Joint Optimization Algorithm
### UAV Position Optimization
The constraint (31) can be approximated as follows:  
\[l_{k}^{(l)}+\mathbf{l}_{k}^{(l)}\,{}^{\mathrm{H}}\Big{(}\mathbf{q}-\mathbf{q}^{(l)}\Big{)} \geq 0,\forall k \tag{32}\]  
where  
\[l_{k}^{(l)} =\eta\left(\mathbf{W}_{k}^{\epsilon},d(\mathbf{q}^{(l)},\mathbf{u}_{k})\right) \tag{33}\] \[-\Gamma_{k}^{\mathrm{com}}\left(\sum\limits_{i=1,i\
eq k}^{K}\eta \left(\mathbf{W}_{i}^{\epsilon},d(\mathbf{q}^{(l)},\mathbf{u}_{k})\right)\,+\sum\limits_{ n=1}^{N}\mu\Big{(}\,\mathbf{W}_{n}^{\epsilon},d(\mathbf{q}^{(l)},\mathbf{u}_{k})\Big{)}\,+ \frac{\sigma_{k}^{2}}{\beta}d^{2}(\mathbf{q}^{(l)},\mathbf{u}_{k})\right)\] \[\mathbf{I}_{k}^{(l)} =\mathbf{\phi}\left(\mathbf{W}_{k}^{\epsilon},d\Big{(}\mathbf{q}^{(l)},\mathbf{ u}_{k}\Big{)}\right)\] (34) \[-\Gamma_{k}^{\mathrm{com}}\left(\sum\limits_{i=1,i\
eq k}^{K}\bm {\phi}\Big{(}\mathbf{W}_{i}^{\epsilon},d\Big{(}\mathbf{q}^{(l)},\mathbf{u}_{k}\Big{)} \Big{)}\,+\sum\limits_{n=1}^{N}\delta\Big{(}\mathbf{W}_{n}^{\epsilon},d\Big{(} \mathbf{q}^{(l)},\mathbf{u}_{k}\Big{)}\Big{)}\,+2\frac{\sigma_{k}^{2}}{\beta}\Big{(} \mathbf{q}^{(l)}-\mathbf{u}_{k}\Big{)}\right)\]  
3. The inequality constraint C13 can be reformulated as:  
\[\sum\limits_{\alpha=1}^{N}\left[\mathbf{G}\right]_{\alpha,\alpha}+2\sum\limits_{ a=1}^{N}\sum\limits_{b=a+1}^{N}\big{|}[\mathbf{G}]_{a,b}\big{|}\cos\left(\theta_{a,b}^{ \epsilon}+\pi(b-a)\frac{H}{d(\mathbf{q},\mathbf{e}_{j})}\,\right)\geq\Gamma_{\epsilon }^{\mathrm{sen}}d^{2}(\mathbf{q},\mathbf{e}_{j}) \tag{35}\]  
Similarly, by conducting a first-order Taylor expansion at the local point \(\mathbf{q}^{(l)}\), the inequality constraint is approximated as follows:  
\[U_{j}^{(l)}+\mathbf{V}_{j}^{(l)}\,{}^{\mathrm{H}}\Big{(}\mathbf{q}-\mathbf{q}^{(l)}\Big{)} \geq\Gamma_{\epsilon}^{\mathrm{sen}}\Big{(}H^{2}+\big{\|}\mathbf{q}-\mathbf{e}_{j} \big{\|}^{2}\Big{)} \tag{36}\]  
where  
\[U_{j}^{(l)}=\eta\Big{(}\mathbf{G},d\Big{(}\mathbf{q}^{(l)},\mathbf{e}_{j}\Big{)}\Big{)} \tag{37}\]  
\[-\Gamma_{k}^{\mathrm{com}}\left(\sum\limits_{i=1,i\
eq k}^{K}\eta \left(\mathbf{W}_{i}^{\epsilon},d(\mathbf{q},\mathbf{u}_{k})\right)\,+\sum\limits_{n=1}^{ N}\mu\big{(}\mathbf{W}_{n}^{\epsilon},d(\mathbf{q},\mathbf{u}_{k})\big{)}\,+\frac{\sigma_{k}^{2} }{\beta}d^{2}(\mathbf{q},\mathbf{u}_{k})\right)\geq 0\]0  
Based on the transformation of the above three constraints, the three inequality constraints C11, C12, and C13 of problem P6 are approximated as (23), (32), and (36), respectively. To ensure the accuracy of the SCA algorithm, we consider that the change in position between two consecutive iterations is less than \(R^{(l)}\), i.e., \(\left\|\mathbf{q}^{(l)}-\mathbf{q}^{(l-1)}\right\|\leq R^{(l)}\). Building on the aforementioned approach, problem P6 in the \(l\)-th iteration can be transformed into an approximate convex problem P7.\(l\):  
\[ \begin{split}\text{P7.1}&:\min_{\mathbf{q},B}\\ \text{s.t.C14}&:C_{j,k}^{(l)}+\mathbf{D}_{jk}^{(l)} \operatorname{H}\left(\mathbf{q}-\mathbf{q}^{(l)}\right)\leq\log_{2}(B),\forall j,k\\ \text{C15}&:\mathbf{I}_{k}^{(l)}+\mathbf{I}_{k}^{(l)} \operatorname{H}\left(\mathbf{q}-\mathbf{q}^{(l)}\right)\geq 0,\forall k\\ \text{C16}&:\mathbf{U}_{j}^{(l)}+\mathbf{V}_{j}^{(l)} \operatorname{H}\left(\mathbf{q}-\mathbf{q}^{(l)}\right)\geq\Gamma_{\ell}^{\text{sen} }\Big{(}H^{2}+\left\|\mathbf{q}-\mathbf{e}_{j}\right\|^{2}\Big{)},\forall j\\ \text{C17}&:\ \left\|\mathbf{q}-\mathbf{q}^{(l-1)}\right\| \leq R^{(l)}\end{split}  \tag{39}\]  
Theoretically, setting the trust region radius \(R^{(l)}\) sufficiently small can always ensure the convergence of the iterative process [11]. In practice terms, if the solution of problem P7.\(l\) after a certain iteration does not reduce the objective value of problem P5 compared to the previous iteration, the trust region radius is halved, i.e., \(R^{(l)}=R^{(l)}/2\), and then problem P7.\(l\) is solved again. If the trust region radius \(R^{(l)}\) decreases below a predetermined threshold \(R_{\text{min}}\), the iterative process stops. The detailed procedure is outlined in Algorithm 2.