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In finite dimensions, if the graph is locally closed at $(\bar{u},\bar{v})$, the Aubin property holds exactly when $D^{\ast }S(\bar{u}\mid \bar{v})(0)=\{0\}$. Thus no nonzero horizontal vector in the limiting normal cone may occur. The exact local Lipschitz modulus is the outer norm of this coderivative.

Normal cones describe generalized supporting directions to sets. For a convex set, the normal condition is $⟨w,z^{\prime }-z⟩\le 0$ for every $z^{\prime }$ in the set. For nonconvex sets, the Fréchet normal cone and the limiting normal cone distinguish local regular support from limits of such supports.

Work in finite-dimensional Euclidean spaces. Suppose the graph of a set-valued mapping,is locally closed at $(\bar{u},\bar{v})\in \mathrm{gph}S$. 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 byAt 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:It can be nonconvex. The Mordukhovich coderivative isThe minus sign in the output-dual component is part of the definition. For a single-valued continuously differentiable mapping $s$, the graph normal is $(Ds(\bar{u}{)}^T{v}^{\ast },-v^{\ast })$, giving $D^{\ast }s(\bar{u}\mid s(\bar{u}))(v^{\ast })=\{Ds(\bar{u}{)}^T{v}^{\ast }\}$. Thus the coderivative extends a transpose derivative, rather than the ordinary forward derivative.

The exact Mordukhovich criterion isThis is equivalent to excluding a nonzero horizontal limiting graph normal $(u^{\ast },0)$. 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 normThe 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.