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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?

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    use that $$\sqrt{\frac{a^2+b^2}{2}}\geq \frac{a+b}{2}$$ and use this for $n$ variables – 2017-01-24

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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?