Minimize sum of $w_i(y_i-y)$ given $|y_i-y| \leq 4w_i$
Write
$$ L = \sum_{i=1}^n w_i y_i - \sum_{i=1}^n w_i y. $$
Thus $L$ is linear in $y$, and moreover, it is a decreasing function. Thus we want to choose the largest $y$ which satisfies the constraints $|y_i - y| \leq 4w_i$, if any. I'll let you figure out how to find this $y$.