Karush-Kuhn-Tucker conditions for the Lasso ($X^{\mathsf{T}}(Y-Xb)/n\in \lambda \partial \Vert b{\Vert }_1$)

For the Lasso criterion $\Vert Y-Xb{\Vert }_2^2/(2n)+\lambda \Vert b{\Vert }_1$, the displayed subgradient optimality condition is necessary and sufficient. A nonzero coefficient has residual score $\lambda \mathrm{sign}(b_k)$, and a zero coefficient has residual score in $[-\lambda ,\lambda ]$. The subdifferential of the L1 norm supplies both cases.