I am struggling to solve the following optimization problem
\begin{aligned} \min_{\{u_{1},u_{2},u_{3}\}} & \quad10u_{1}+15u_{2}+5u_{3} \\ \text{s.t.} \quad & \frac{1}{u_{1}}+\frac{1}{u_{2}}+ \frac{1}{u_{3}}= 16 \\ & 0\leq u_{1}\leq \frac{1}{5} ,\;\;0\leq u_{2}\leq \frac{1}{3},\;\;0\leq u_{3}\leq \frac{1}{7}\end{aligned}
which is equivalent to
\begin{aligned} \min_{\{u_{1},u_{2},u_{3}\}} & \quad10u_{1}+15u_{2}+5u_{3} \\ \text{s.t.} \quad & u_{2}u_{3}+u_{1}u_{3}+u_{1}u_{2}= 16 u_{1}u_{2}u_{3} \\ & 0\leq 5u_{1}\leq 1 ,\;\;0\leq 3u_{2}\leq 1,\;\;0\leq 7u_{3}\leq 1\end{aligned}
How can I solve this problem analytically, and what is the type of this optimization problem?