I have an optimization problem with the following constraint:
$$|f(x)| - y \: \Re \{ g(x) \} < 0$$
where $x,y$ are optimization variables (with $x \in \mathbb{R}^n$, $\{y \in \mathbb{R}: y > 0\}$), and where $|f(x)|$ is a convex function, and $g(x)$ is a (complex) linear function of $x$ (with $\Re \{\cdot \}$ denoting the real part).
The term $y \: \Re \{ g(x) \}$ is bilinear, so the problem is not convex. I wanted to know if there was a way to convexify this term. I have tried linearizing this term (first-order Taylor expansion), but I'm not sure if it is appropriate to do this operation on this kind of problem.
Your help is much appreciated! Thanks!