TITLE: How to minimize the square of sum given fixed sum of squares?
QUESTION [0 upvotes]: 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!

REPLY [0 votes]: 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.
{"set_name": "stack_exchange", "score": 0, "question_id": 4303653}