Algorithm 1 SQP for Antenna Position Optimization

1:Initial: position ${\mathbf{\chi }}_0={[{\mathbf{x}}_0^T,{\mathbf{z}}_0^T]}^T$, hessian approximation ${\mathbf{U}}_0={\mathbf{I}}_{2N\times 2N}$, lagrange multipliers ${\mathbf{\lambda }}_0=\mathbf{0}$, tolerance $\mathit{ϵ}>0$, penalty parameter $\mathit{\mu }>0$ 2:Set $t\leftarrow 0$ 3:while$\parallel {\mathbf{\chi }}_{t+1}-{\mathbf{\chi }}_t{\parallel }_2 \ge \mathit{ϵ}$or$max{g}_{ij}\left({\mathbf{\chi }}_{t+1}\right)>\mathit{ϵ}$do 4:    Compute $\nabla f\left({\mathbf{\chi }}_t\right)$ and $\nabla g_{ij}\left({\mathbf{\chi }}_t\right)$ 5:     Solve the QP: $$\begin{array}{cc}\hfill \underset{\mathbf{d}}{min}& {\displaystyle \frac{1}{2}}{\mathbf{d}}^T{\mathbf{U}}_t\mathbf{d}+\nabla f{\left({\mathbf{\chi }}_t\right)}^T\mathbf{d}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \nabla g_{ij}{\left({\mathbf{\chi }}_t\right)}^T\mathbf{d}\le -g_{ij}\left({\mathbf{\chi }}_t\right),\forall i<j\hfill \end{array}$$ 6:    Obtain search direction ${\mathbf{d}}_t$ and multipliers ${\mathbf{\lambda }}_{t+1}$ 7:     Find step size ${\mathit{\alpha }}_t\in (0,1]$ minimizing the merit function: $$\begin{array}{c}\hfill \mathit{\varphi }(\mathit{\alpha })=f({\mathbf{\chi }}_t+\mathit{\alpha }{\mathbf{d}}_t)+\mathit{\mu }\sum _{i<j}max\left(0,g_{ij}({\mathbf{\chi }}_t+\mathit{\alpha }{\mathbf{d}}_t)\right)\end{array}$$ 8:    ${\mathbf{\chi }}_{t+1}\leftarrow {\mathbf{\chi }}_t+{\mathit{\alpha }}_t{\mathbf{d}}_t$ 9:    ${\mathbf{m}}_t={\mathbf{\chi }}_{t+1}-{\mathbf{\chi }}_t$ 10:   ${\mathbf{r}}_t\leftarrow {\nabla }_{\mathbf{\chi }}\mathcal{L}({\mathbf{\chi }}_{t+1},{\mathbf{\lambda }}_{t+1})-{\nabla }_{\mathbf{\chi }}\mathcal{L}({\mathbf{\chi }}_t,{\mathbf{\lambda }}_{t+1})$ 11:    Update ${\mathbf{U}}_{t+1}$ via BFGS: $$\begin{array}{c}\hfill {\mathbf{U}}_{t+1}={\mathbf{U}}_t-{\displaystyle \frac{{\mathbf{U}}_t{\mathbf{m}}_t{\mathbf{m}}_t^T{\mathbf{U}}_t}{{\mathbf{m}}_t^T{\mathbf{U}}_t{\mathbf{m}}_t}}+{\displaystyle \frac{{\mathbf{r}}_t{\mathbf{r}}_t^T}{{\mathbf{m}}_t^T{\mathbf{r}}_t}}\end{array}$$ 12:    $t\leftarrow t+1$ 13:end while 14:Return: ${\mathbf{\chi }}^{\ast }\leftarrow {\mathbf{\chi }}_t$