I have a problem that's insisting that I formulate a function $f(\vec x) \to \vec y$, where $\vec x \in \mathbb R^{2n}$ and $\vec y \in \mathbb C^n$.
When I take the Jacobian of $f$, I end up with a $2n \times n$ matrix of complex numbers. This is distressing, because I'm doing an optimization problem that really "wants" a full-rank Jacobian. On the other hand, each complex number carries two real ones, so there's got to be a full rank matrix in there somehow.
Based on the way the problem is structured, I'm pretty sure that I can declare a "real valued" Jacobian $J_r = \begin{bmatrix}\mathcal Re{\nabla f(x)} \\ \mathcal Im{\nabla f(x)}\end{bmatrix}$ and then use it as if it were arrived at by perfectly normal means.
Is there a way to treat this that's either more theoretically robust, more compact, or both? Or am I already doing this correctly, and this is the best I get?