The best w_j confusion in xgboost
Usually to find an optimum you set the derivative of the function equal to 0. In your case that gives

$$ \frac{dG_j}{dw_J} = G_j + (H_j+\lambda)w_j = 0 $$

leading to 

$$ w_j = \frac{-G_j}{H_j+\lambda}$$