Let $A$ be positive-definite and $C$ diagonal positive-definite, consider the problem of solving the following equation for $\bf x$ $$A{\bf x}+C\begin{bmatrix} e^{x_1} \\ \vdots \\ e^{x_n} \end{bmatrix}=b.$$
I believe nonlinear conjugate gradient can solve this, however the nonlinear part is pretty benign as far as nonlinearities go, is there a trick to put this into a form where conjugate gradient could solve it?