Given a set of constants $a_1,\ldots,a_n$, I want to solve the following single-variable optimization problem:
$$\min_x \sum_{i=1}^n \frac{a_i^2}{x(2a_i-x)}, \quad s.t. \quad 0\leq x \leq 2a_i, \forall i$$
I initially thought it is only a single-variable problem so should be easy to solve... I set the derivative to zero and got to the point where $$ \sum_{i=1}^n\frac{a_i^2}{x^2(2a_i-x)} = \sum_{i=1}^n\frac{a_i^2}{x(2a_i-x)^2}$$ then got stuck.
I also tried a few instances on https://www.wolframalpha.com and saw that a minimizer seems to exist.
Any insight into solving the problem is highly appreciated. Thanks in advance!