Variable projection

In variable projection, a prediction has the form $\mathbf{G}(\mathbf{q})\mathbf{m}$, nonlinear in $\mathbf{q}$ but linear in $\mathbf{m}$. For each trial $\mathbf{q}$, solve the inner weighted least squares problem for $\mathbf{m}$, then optimize the resulting residual over $\mathbf{q}$. This reduces the nonlinear dimension without pretending that the whole inverse problem is linear.