= Solution
Work in finite-dimensional Euclidean spaces. Suppose the <graph of a set-valued mapping>,
$$
\operatorname{gph}S=\{(u,v):v\in S(u)\},
$$
is locally closed at $(\bar u,\bar v)\in\operatorname{gph}S$. Local closedness means that its intersection with some neighborhood of this point is closed relative to that neighborhood. This assumption is part of the criterion.
For a set $C$ and $z\in C$, define the <Fréchet normal cone> by
$$
\widehat N_C(z)=\left\{w:\limsup_{\substack{z'\to z\\z'\in C,\ z'\ne z}}
\frac{\langle w,z'-z\rangle}{\|z'-z\|_2}\leq0\right\}.
$$
At an isolated point the condition is vacuous, so all vectors are regular normals. The <limiting normal cone> consists of limits of these regular normals at nearby points:
$$
N_C(z)=\left\{w:\exists z_k\in C,\ z_k\to z,\ \exists w_k\in\widehat N_C(z_k),\ w_k\to w\right\}.
$$
It can be nonconvex. The <Mordukhovich coderivative> is
$$
\boxed{D^*S(\bar u\mid\bar v)(v^*)
=\{u^*:(u^*,-v^*)\in N_{\operatorname{gph}S}(\bar u,\bar v)\}}.
$$
The minus sign in the output-dual component is part of the definition. For a single-valued continuously differentiable mapping $s$, the graph normal is $(Ds(\bar u)^Tv^*,-v^*)$, giving $D^*s(\bar u\mid s(\bar u))(v^*)=\{Ds(\bar u)^Tv^*\}$. Thus the <coderivative> extends a transpose derivative, rather than the ordinary forward derivative.
The exact <Mordukhovich criterion> is
$$
\boxed{S\text{ has the Aubin property at }(\bar u,\bar v)
\iff D^*S(\bar u\mid\bar v)(0)=\{0\}}.
$$
This is equivalent to excluding a nonzero horizontal limiting graph normal $(u^*,0)$. One must use the <limiting normal cone>; simply checking regular normals at the reference point can miss normals inherited from neighboring graph pieces.
For <sensitivity analysis>, apply this test to a solution map or use <coderivative> and <normal cone> calculus to express its graph normals through optimality constraints. Triviality of this kernel then proves <Lipschitz-like> stability without solving the perturbed problem explicitly. Under these same finite-dimensional, locally closed assumptions, the exact Lipschitz modulus is the outer norm
$$
\operatorname{lip}S(\bar u\mid\bar v)
=\sup\{\|u^*\|_2:u^*\in D^*S(\bar u\mid\bar v)(v^*),\ \|v^*\|_2\leq1\}.
$$
The zero-kernel condition is the qualitative criterion; the modulus quantifies the sensitivity bound. For the next problem an explicit local formula is simpler than evaluating these graph normals.
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