Let $f_j:\mathbb{R}^n\rightarrow[0,\infty[$ be a non-negative function, for $j=1,...,p$. It is known that optimization problem $$\min_{x\in\mathbb{R}^n}\;\sum_{j=1}^p f_j(x)$$ has a solution given by $\widehat{x}\in\mathbb{R}^n$. Can I said that a solution of the optimization problem $$\min_{x\in\mathbb{R}^n}\;\sum_{j=1}^p (f_j(x))^2$$ is given by $\widehat{x}$, as well?
If answer is Yes, could you give me some reference of this result?
Let $f_1(x) = \frac{1}{2} (x-10)^2$ and $f_2(x) = (x - 100)^2$. The solution to the first problem is
$$0 = x-10 + 2x - 200\implies x=70$$
meanwhile for the second problem we have
$$0 = (x-10)^2(x-10) + 2(x-100)^2(2x-200)\implies x = 82 - 36 2^{1/3} + 18\times 2^{2/3}$$