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For a sample term $\max \{0,1-a_{i}x\}$, the active hinge set contains indices with $1-a_{i}x>0$. At strict negative margins the hinge is inactive, while zero margins allow fractional hinge-loss optimality weights. Away from zero margins, the active set stays fixed on a sufficiently small candidate solution neighborhood.

Write the objective as $F_a(x)=\alpha x^2/2+n^{-1}{\sum }_i{\ell }_i(x)$ with ${\ell }_i(x)=\max \{0,1-a_{i}x\}$. Each hinge loss is finite and convex, so the subdifferential sum rule applies without a qualification gap. Its subdifferential isThis formula also covers $a_i=0$, if such a sample is allowed, because the loss is then constant. For a convex function, zero in the subdifferential is necessary and sufficient for global optimality. Thus $x$ is optimal exactly when there are weights ${\lambda }_i$ such thatEquivalently, using $I(x)=\{i:a_{i}x<1\}$ and $J(x)=\{i:a_{i}x=1\}$,The notation $[0,1]a_i$ denotes the segment between $0$ and $a_i$, also when $a_i<0$. This avoids reversing interval endpoints for negative-class samples.

Because $\alpha >0$, the objective is strongly convex, and $F_a(x)\ge \alpha x^2/2$ makes it coercive. It is continuous, so a minimizer exists and is unique for every finite data vector $a$. The weights at exact margins need not be unique even though $x$ is. These are the hinge-loss optimality weights for the one-dimensional support vector machine.

Yes: the solution map is locally affine, hence has the Aubin property. Keep the regularization parameter $\alpha >0$ fixed and define the active hinge setSince none of the margins vanishes, the optimality condition at the reference data has no fractional weights:For nearby $a$, define the candidateEach map $a\mapsto 1-a_i{x}_I(a)$ is continuous and is nonzero at $\hat{a}$. Because there are finitely many samples, all their signs remain unchanged on a common neighborhood $U$ of $\hat{a}$. Therefore the same set $I$ is active at $x_I(a)$ for every $a\in U$. The hinge-loss optimality weights are still $1$ on $I$ and $0$ off $I$, and the defining equation for $x_I(a)$ proves the subgradient optimality condition. Strong convexity makes this candidate the unique global minimizer. Consequently the solution map of a parametric optimization problem satisfiesThis argument establishes stability of the active pattern without presupposing continuity of the unknown optimizer; continuity is used only for an explicit candidate and then optimality is checked.

For $a,b\in U$, the Cauchy-Schwarz inequality givesThis is the Aubin property with $\kappa =\sqrt{|I|}/(\alpha n)$. In fact this constant is the exact local Lipschitz continuity modulus of the affine branch, since its derivative isThe active-set dependence of this derivative is the active-set sensitivity of hinge-loss minimization. At a vanishing margin the affine-branch argument no longer applies, although failure of this particular argument does not by itself prove failure of the Aubin property.

For the illustration, take $n=2$, $\alpha =1$, and $a=(s,1)$ with $s>0$. The hinge-loss optimality weights giveOn the middle branch the first margin is exactly zero, so that branch lies outside the strict-margin hypothesis. The example makes clear why the local affine conclusion is tied to that hypothesis, and also shows that a zero margin need not destroy local Lipschitz stability.