The problem
Find the solution for $argmin_{(\mathbf{u},\mathbf{q},\mathbf{f})} \,\mathcal{J}$, and $argmax_{(\mathbf{u},\mathbf{q},\mathbf{f})} \,\mathcal{K}$, with,
$\mathcal{J}=||\mathbf{I} \, \mathbf{f}-\mathbf{y}||^2$
$\mathcal{K}=||\mathbf{q}||^2$.
$\mathbf{u}$ is linked to $\mathbf{f},\mathbf{q}$ by the non-linear equation $\mathcal{N}$,
$\mathcal{N}(\mathbf{f},\mathbf{q},\mathbf{u})=0$.
$\mathbf{f},\mathbf{q},\mathbf{u}$ are all unknown discrete complex vectors of size $n$.
$\mathbf{y}$ is a known complex vector of size $m$, with $m<n$. $\mathbf{I}$ is a known matrix of size $n\times m$.
$\mathcal{N}$ is differentiable, $\partial_\mathbf{x} \, \mathcal{N}$ is known for all variables $\mathbf{x}$
Some context
This is actually the inverse problem of what I usually solve, where $\mathbf{u}$ is known but not $\mathbf{y}$. If $\mathbf{u}$ is known, $\mathbf{f}$ can easily be found using non-linear optimization and the adjoint equation to $\mathcal{N}$.
I feel like it should be solvable using some weird successive non-linear iterative approach. Reasonable guess value can be provided for both $\mathbf{f}$ and $\mathbf{u}$. I have tried to reduce the problem to its essence, I can give more detail if necessary.
Would it be reasonable to optimize the following Lagrange function?
$\mathcal{L}(\mathbf{u}^{\dagger},\mathbf{u},\mathbf{q},\mathbf{f})= \, + \,\mathcal{J}(\mathbf{f}) \, - \,\mathcal{K}(\mathbf{q}) \,\,\,- \,< \mathbf{u}^{\dagger},\mathcal{N}(\mathbf{u},\mathbf{q},\mathbf{f})>$
Note the different signs before $\mathcal{K}$ and $\mathcal{J}$.