# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 3 Joint Optimization Algorithm
### UAV Position Optimization
\[C_{j,k}^{(l)} =\log_{2}\Big{(}\eta\left(\mathbf{W}_{k}^{c},d\big{(}\mathbf{q}^{(l)},\bm {e}_{j}\big{)}\right)\Big{)}\] (24) \[D_{j,k}^{(l)} =\frac{\log_{2}(e)}{f_{j,k}}\Big{(}\mathbf{\phi}\big{(}\mathbf{W}_{k}^{c},d\Big{(}\mathbf{q}^{(l)},\mathbf{e}_{j}\big{)}\big{)}\Big{)}\] (25) \[\phi\Big{(}\mathbf{W}_{k}^{c},d\Big{(}\mathbf{q}^{(l)},\mathbf{e}_{j}\big{)} \Big{)} =\sum\limits_{a=1}^{N}\sum\limits_{b=a+1}^{N}2\pi\Big{|}\big{[}\bm {W}_{k}^{c}\big{]}_{a,b}\Big{|}\sin\left(\theta_{a,b}^{\mathbf{W}_{k}^{c}}+\pi(b-a) \frac{H}{d\big{(}\mathbf{q}^{(l)},\mathbf{e}_{j}\big{)}}\right)\frac{H(b-a)}{d^{3} \big{(}\mathbf{q}^{(l)},\mathbf{e}_{j}\big{)}}\] (26) \[-\log_{2}\left(\sum\limits_{i=1,j+k}^{K}\eta\left(\mathbf{W}_{i}^{c},d \big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\right)\right.+\sum\limits_{n=1}^{N}\mu\big{(} \mathbf{W}_{n}^{c},d\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\big{)}\right.+\frac{{\sigma_{e} }^{2}}{\beta}d^{2}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\bigg{)}\leq\log_{2}(B)\]0  
Within the derivative calculation of the logarithmic function, \(f_{j,k}\) and \(g_{j,k}\) are introduced as the denominators, respectively represented as:  
\[-\log_{2}\left(\sum\limits_{i=1,j+k}^{K}\eta\left(\mathbf{W}_{i}^{c},d \big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\right)\right.+\sum\limits_{n=1}^{N}\mu\big{(} \mathbf{W}_{n}^{c},d\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\big{)}\right.+\frac{{\sigma_{e} }^{2}}{\beta}d^{2}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\bigg{)}\leq\log_{2}(B)\]1\[g_{jk}=\sum\limits_{ \begin{subarray}{c}i=1,\\ i\
eq k\end{subarray} }^{K}\eta\Big{(}\,\mathbf{W}_{i}^{\epsilon},d(\mathbf{q}^{(l)},\mathbf{e }_{j})\Big{)}\,+\sum\limits_{n=1}^{N}\mu\Big{(}\,\mathbf{W}_{n}^{\epsilon},d(\mathbf{q} ^{(l)},\mathbf{e}_{j})\Big{)}\,+\frac{\sigma_{\epsilon}^{2}}{\beta}d^{2}(\mathbf{q}^{( l)},\mathbf{e}_{j}) \tag{29}\]  
At this point, the inequality constraint (23) becomes convex regarding the optimization variable \(\mathbf{q}\) and auxiliary variable \(B\);  
2. Convert the inequality constraint C12 into a polynomial form:  
\[\Big{|}\mathbf{a}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega}_{k}^{\epsilon}\Big{|} ^{2}-\Gamma_{k}^{\mathrm{com}}\left(\sum\limits_{ \begin{subarray}{c}i=1,\\ i\
eq k\end{subarray} }^{K}\Big{|}\mathbf{a}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{ \omega}_{i}^{\epsilon}\Big{|}^{2}+\sum\limits_{n=1}^{N}\Big{|}\mathbf{a}^{ \mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega}_{n}^{\gamma}\Big{|}^{2}+\frac{ \sigma_{k}^{2}}{\beta}d^{2}(\mathbf{q},\mathbf{u}_{k})\right)\geq 0 \tag{30}\]  
For simplicity of expression, (30) can be rewritten as:  
\[\eta\left(\mathbf{W}_{k}^{\epsilon},d(\mathbf{q},\mathbf{u}_{k})\right) \tag{31}\] \[-\Gamma_{k}^{\mathrm{com}}\left(\sum\limits_{i=1,i\
eq k}^{K}\eta \left(\mathbf{W}_{i}^{\epsilon},d(\mathbf{q},\mathbf{u}_{k})\right)\,+\sum\limits_{n=1}^{ N}\mu\big{(}\mathbf{W}_{n}^{\epsilon},d(\mathbf{q},\mathbf{u}_{k})\big{)}\,+\frac{\sigma_{k}^{2} }{\beta}d^{2}(\mathbf{q},\mathbf{u}_{k})\right)\geq 0\]  
Similarly, given the non-convex nature of the inequality with respect to the optimization variable \(\mathbf{q}\), the left-hand side is linearized via a first-order Taylor expansion at the local point \(\mathbf{q}^{(l)}\).