help me minimize this sum

Question

I want minimize $\sum_{i} ^{n} w_i^2$ such that $\sum_i ^{n} w_i=1$, and $w_i>0$

simply taking the derivative and set it to zero won't work. The answer is probably $w_i=1/n$, for all $i$. But I don't know how to show that.

Any thought on this?

use that $$\sqrt{\frac{a^2+b^2}{2}}\geq \frac{a+b}{2}$$ and use this for $n$ variables — Dr. Sonnhard Graubner, Jan 24 '17 at 18:53

score 1 · Accepted Answer · answered Jan 24 '17 at 18:58

1

we have $$\sqrt{\frac{1}{n}\sum_{i=1}^{n}a_i^2}\geq \frac{1}{n}\sum_{i=1}^na_i$$ can you proceed?

answered Jan 24 '17 at 18:58

1 Answers1