Minimize $\left(\sum_{i=1}^{n}x_{i}\right)\left(\sum_{i=1}^{n}\frac{a_{i}}{x_{i}}\right)$

Question

Let $x, a\in\mathbb{R}^{n}_{++}$ and define \begin{align} \mathcal{L}(x) = \left(\sum_{i=1}^{n}x_{i}\right)\left(\sum_{i=1}^{n}\frac{a_{i}}{x_{i}}\right). \end{align} Find the minimizer $x$ of $\mathcal{L}$.

Answer 1

By Cauchy-Schwarz $$ \left( \sum_{i=1}^n \sqrt{a_i} \right)^2 = \left( \sum_{i=1}^n \sqrt{x_i} \cdot \sqrt{\frac{a_i}{x_i}} \right)^2 \le \left( \sum_{i=1}^n x_i \right) \cdot \left( \sum_{i=1}^n \frac{a_i}{x_i} \right). $$ The equality in Cauchy-Schwarz takes place iff $x_i^2 = ba_i$ for some $b$.