= Active-set sensitivity of hinge-loss minimization
{title2=$Dx(a)[h]=(\alpha n)^{-1}\sum_{i\in I}h_i$}
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 $\sqrt{|I|}/(\alpha n)$.
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