Levenberg-Marquardt minimization

Given a vector relation ${\tt y}=f({\tt x})$ where ${\tt x}$ and ${\tt y}$ can have different dimensions and an observation ${\tt\hat{y}}$, we want to find the vector ${\tt x}$ which best satisfies the given relation. More precisely, we are looking for the vector ${\tt\hat{x}}$ satisfying ${\tt\hat{y}}=f({\tt\hat{x}}) + {\tt\hat{e}}$ for which $\Vert{\tt\hat{e}}\Vert$ is minimal.