Let $c_1,c_2 \in \mathbb{R}$ be constants. Let $u_1,u_2 \in \mathbb{R}^n$ be vectors.
Consider the function $h: \mathbb{R}^n \to \mathbb{R}$ given by
$$ h(x) = c_1 \exp ( \langle u_1,x \rangle) + c_2 \exp( \langle u_2,x \rangle) $$
I would like to approximate this function with $$ g(x) = c \exp ( \langle u,x \rangle) $$ with $c\in \mathbb{R}$ and $u\in \mathbb{R}^n$.
Is there an easy way to find the, in some appropriate sense, optimal choice of $c$ and $u$?
Edit: Furthermore, say I am particularly interested in the approximation on some region, $A$ to the extent I do not care about $A^c$ whatsoever (an example is $A= \{x \in \mathbb{R}^n : h(x)>0 \}$) is this possible to take into account?