I have the following problem whose optimal solution (if possible), I would like to find.
$\min_{\mathbf{f}} \left\| \mathbf{L}_1 \mathbf{f} \right\|^2_2 + \left\| \mathbf{L}_2 \mathbf{f} \right\|^2_2 $
$\text{s.t} \\ \left| \mathbf{r}_1^H \mathbf{L}_1 \mathbf{f} \right|^2 \geq \left| \mathbf{r}_1^H \mathbf{L}_2 \mathbf{f} \right|^2 + q_1 \\ \left| \mathbf{r}_2^H \mathbf{L}_1 \mathbf{f} \right|^2 \geq \left| \mathbf{r}_2^H \mathbf{L}_2 \mathbf{f} \right|^2 + q_2 \\ \mathbf{f} = \exp(j\boldsymbol{\theta}) $
where $\mathbf{L}_1, \mathbf{L}_2, \mathbf{r}_1, \mathbf{r}_2$ are complex matrices and vectors. $\boldsymbol{\theta}$ is a vector.
To the best of my knowledge, because the unknown variable $\boldsymbol{\theta}$ is embedded into an exponential function, the objective is non-convex. The same applies to the constraints. Otherwise, if there were no restriction on $\mathbf{f}$, Slater condition would have guaranteed strong duality allowing to solve the problem optimally. Maybe I am not aware of other regularity conditions that could help me solve the problem in an optimal manner. If not possible to attain optimality, would you please suggest any suboptimal alternatives. Thank you.