I thought of writing this question Minimum for this function in a different way, if it helps.
I want to minimize
$$\sum_{i=1}^n a_ix_i + \nu \sum_{i=1}^n b_i 2^{x_i} ,$$
where $a_i \in [0,1]$, $b_i \in (0,\infty)$, and $x_i \in [x_{\min},0)$, and
$$ \sum_{i=1}^n 2^{x_i} = 1 .$$
$x_{\min}$, $\nu$, $a_i$ and $b_i$ are constants.
I guess the tricky part is to minimize the function while ensuring all $2^{x_i}$ sum up to $1$ for these $n$ variables.
Thank you for your patience and help.