Active-set sensitivity of hinge-loss minimization ($Dx(a)[h]=(\alpha n{)}^{-1}{\sum }_{i\in I}{h}_i$)

With fixed positive quadratic regularization, if all reference hinge margins are nonzero, the one-dimensional minimizer has local formula $x(a)=(\alpha n{)}^{-1}{\sum }_{i\in I}{a}_i$. The active hinge set is verified using continuity of this explicit candidate, then strong convexity identifies it as the optimizer. Its exact Euclidean Lipschitz modulus is $\sqrt{|I|}/(\alpha n)$.