 Solution

For hinge loss, chooseThe subgradient optimality condition isInjectivity of $X^T$ gives $t/n=2\lambda \hat{\alpha }$. Multiplying by $X{X}^T$ and writing $k_i$ for its $i$th column yieldsIf $y_i{x}_i^T\hat{\beta }>1$, then $t_i=0$, and $t/n=2\lambda \hat{\alpha }$ implies ${\hat{\alpha }}_i=0$.