 Solution

Let $S:{\mathbb{R}}^p\rightrightarrows {\mathbb{R}}^q$ be a set-valued mapping and let $\bar{v}\in S(\bar{u})$. The Aubin property at $(\bar{u},\bar{v})$ means there are neighborhoods $U$ of $\bar{u}$, $V$ of $\bar{v}$, and a finite constant $\kappa \ge 0$ such thatHere ${\mathbb{B}}_q$ is the closed unit ball. Equivalently, every solution $v\in S(u)$ near $\bar{v}$ can be matched to a solution $v^{\prime }\in S(u^{\prime })$ within $\kappa \Vert u-u^{\prime }{\Vert }_2$. The localization is on the left-hand side: the matching point need not be in $V$. The property is also called the Lipschitz-like property.

For sensitivity analysis, let $u$ represent data or perturbations and $S(u)$ the set of feasible or optimal solutions. The Aubin property bounds how far a nearby solution can move when the data change. Taking $u=\bar{u}$, $v=\bar{v}$ also guarantees a nearby solution for each sufficiently small perturbation $u^{\prime }$. It is a stability estimate for a relation, and by itself does not imply uniqueness or differentiability. For a single-valued solution map, it reduces to local Lipschitz continuity.