Consider the following maximization problem:
$$\max_{(x_1,\dots,x_n)\in \mathbb{R}^n}{f(\mathbf{x})=\prod_{i=1}^n x_i^{\alpha_i}}$$
subject to
$$ \sum_{i=1}^n x_i = c; \ 0\leq a_i \leq x_i \leq b_i \ \forall i \in 1,\dots, n$$
I'm not sure the right term for $f$ is multinomial: I guess that would be the correct nomenclature of all $\alpha_i$ were integer, but that's not necessarily the case. Anyway, I'd like to solve this optimization problem. My first idea was to take the log of $f$, in order to deal with a linear objective function. However, that would mess up with the first constraint (the others constraints would be fine). So what could be the best approach? Is there some specific algorithm I could use?