Hinge-loss optimality weight (${\lambda }_i\in [0,1]$)

The subgradient of $\max \{0,1-a_{i}x\}$ is $-{\lambda }_i{a}_i$, with ${\lambda }_i=1$ below the margin, ${\lambda }_i=0$ beyond it and ${\lambda }_i\in [0,1]$ at equality. For quadratic regularization the one-dimensional optimality condition is $\alpha nx={\sum }_i{\lambda }_i{a}_i$. The optimizer can be unique even if the weights at equality are not.