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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?
The trick is to view your equation as the first-order optimality conditions of an unconstrained optimization problem $\min_x f(x)$. Recall that the first-order optimality conditions are just $\nabla f(x) = 0$. So in your case, you only need to set $$ f(x) = -b^T x + \tfrac{1}{2} x^T A x + \sum_{i=1}^n C_{ii} e^{x_i}, $$ where I'm assuming $A$ is symmetric (you didn't specify) and $C_{ii}$ is the $i$-th diagonal element of $C$.
