Given the real values $\{f_1,\ldots,f_m\}$ and $\{\theta_1,\ldots,\theta_d\}$, I want to solve for $x_k$ and $y_k$ for $k\in\{1,\ldots,n\}$ this problem:
\begin{align}\text{maximize} \quad& \left|\sum_{k=1}^ne^{j2\pi fx_k}e^{jy_k}e^{-jk\alpha(\theta)}\right|^2,\\\text{subject to } \quad & \alpha(\theta)=\pi \sin(\theta_i) \text{ for all } f\in\{f_1,\ldots,f_m\} \text{ and } i\in\{1,\ldots,d\}.\end{align}
where $j=\sqrt{-1}$.
This translates into $m$ optimization problems as follows, where problem $p$ is given by: \begin{align} \text{maximize } & \left|\sum_{k=1}^ne^{j2\pi f_px_k}e^{jy_k}e^{-jk\alpha(\theta_i)}\right|^2,\\ \text{subject to } & \alpha(\theta_i)=\pi \sin(\theta_i),\text{ for } i\in\{1,\ldots,d\}.\end{align}
What is the best way to solve this optimization problem? Can we solve it using mathematical programming solvers?
Writing down the absolute value squared of the objective function, I find that $$\left|\sum_{k=1}^ne^{j2\pi fx_k}e^{jy_k}e^{-jk\alpha(\theta)}\right|^2=n+2\sum_{1\leq k<k'\leq n}\cos(\Phi_k-\Phi_{k'}),$$ where $\Phi_k=2\pi fx_k+y_k-k\alpha(\theta).$ What to do now?