Is the optimum of this problem unique? I have a convex optimisation problem:
$$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ 
The KKT conditions are:
 $$\lambda=\begin{cases}
a_i/b_i \exp(- x_i/b_i)  &\text{ if }a_i/b_i>\lambda \\
a_i/b_i+\mu_i &\text{ else }\end{cases},$$
where $\lambda>0$ and $\mu_i\ge 0$.
I am able to characterise the interior solution ($x_i>0\ \forall i$), but I am unable to see if the solution is unique and under what circumstances. What is the unique optimum for this problem and how is it dependent on the parameters?

Hint: A strictly convex function has at most one (global) minimum. 
