[Math] Minimization of log-sum-exponential function subject to constraints.

Question

I would like to minimize the following function:

$f(x)=log(e^{-x_1}+..+e^{-x_n})$

Subject to:

$\sum_{i=1}^{n}{x_i}=1$

$0 \leq x_i \leq 1$

So far I have discovered the following: If all the $x_i$'s are equal, $f(x)=max(x_i)+log(n)$, but I have not been able to find a conclusive answer regarding what would be the value of the $x_i$'s for the minimum value of the function.

Answer 1

log is increasing, so the minimum occurs at the same place as the minimum of $\sum e^{-x_i}$; and $x\mapsto e^{-x}$ is convex, so $\sum e^{-x_i} \ge ne^{-\frac1n\sum x_i} = ne^{-1/n}$, with equality when all $x_i$ are equal.