Minimizing sum of $w_i(y_i - y_c)$ over $y_c$
$x_i$ is irrelevant, it doesn't appear in the objective function.
$y_i$ and $w_i$ are constant.
So the minimization simplifies to:
$$\operatorname*{argmin}_{y_c} \sum{i=1}^n w_i(y_i - y_c) = \operatorname*{argmin}_{y_c} \sum_{i=1}^n w_iy_i - \sum_{i=1}^n w_iy_c,$$
which is either
$$\operatorname*{argmin}_{y_c} -\left(\sum_{i=1}^n w_i\right) y_c $$
or
$$\operatorname*{argmax}_{y_c} \left(\sum_{i=1}^n w_i\right) y_c $$
So depending on whether the sum of your weights is positive or negative, either the highest or lowest $c_i$ should be $c$.