How to minimize the square of sum given fixed sum of squares?

Question

This looks simple but I can't find it:

$\begin{aligned} &\min \frac{1}{N}\left(\sum_{i=1}^{N} w_{i}\right)^{2} \\ &\text { s.t. }\|\mathbf{w}\|_{2}^{2}=N \end{aligned}$

Thanks!

Answer 1

Assuming all $w_{i}\geq 0$ then your objective function is increasing so the solution to your problem will be the same as

$min \quad \sum_{i=1}^{N}w_{i}\quad s.t.\quad \sum_{i=1}^{N}w_{i}^{2}=N^{2}$

First assume that $w_{i}=w$ for all $i$, then $w_{i}=\sqrt{N}$ and $\sum_{i=1}^{N}w_{i}=N^{\frac{3}{2}}$.

Now suppose that $w_{1}>0$ and $w_{i}=0$ for $i\geq 2$, then $w_{1}=N$ and $\sum_{i=1}^{N}w_{i}=N$. This will be our solution.