 Solution

The objective separates by coordinates, so minimizeThe subgradient optimality condition isFor $z>0$ this gives $z=x_i-\lambda$, valid when $x_i>\lambda$; for $z<0$ it gives $z=x_i+\lambda$, valid when $x_i<-\lambda$. At $z=0$, the condition is $x_i\in [-\lambda ,\lambda ]$. Therefore the shrinkage operator is the soft-thresholding operator