Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 326 / 2 / c

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

The graph is continuous and piecewise linear: it is horizontal at zero on $[-\lambda ,\lambda ]$ and has slope one outside that interval, joining the axis at $(-\lambda ,0)$ and $(\lambda ,0)$.

 psi
  ^
  |                    /
  |                   /
--+---------==========--------> x
 /         -lambda  lambda
/