Given $n$ functions, $f_i$ $(i=1,\dots,n)$ all of which are in the same function form of $\vec{x}$ but different coefficients in the exponent
$$f_i(\vec{x}) = \exp \left( \sum_{j=1}^m c_{ij} x_j \right)$$
Is there a way to determine the global minimum of
$$F=\sum_{i=1}^N f_i(\vec{x})$$
In an efficient manner for large number of unknowns? It seems that there is a lot of symmetry which could be exploited, especially since $F$ is sum of similar expressions. (In our problem set-up there always exists an unique global solution)
Illustrative Example: $$f_1 = \exp(+.5 x_0 - 2 x_1) \\ f_2 = \exp(+.5 x_0 - 1 x_1) \\ f_3 = \exp(-.5 x_0 + 5 x_1)\\ f_4 = \exp(-.5 x_0 - 5 x_1)$$
With $F=f_1+f_2+f_3+f_4$ has global minimum at $\vec{x}=\{0.137532, 0.0612319\}$.
Hoping there might be some symmetry tricks, or iterative procedures than can be used; currently for small scale problems using Newton's Method for numerical solving which produces $m*m$ matrix for Hessians.