Abstract

Let $\Gamma:=\{x\in \mathbb{R}^n\, |\, q(x)\in\Theta\},$ where $q: \mathbb{R}^n\rightarrow\mathbb{R}^m$ is a twice continuously differentiable mapping, and $\Theta$ is a nonempty polyhedral convex set in $\mathbb{R}^m.$ In this paper, we first establish a formula for exactly computing the graphical derivative of the normal cone mapping $N_\Gamma:\mathbb{R}^n\rightrightarrows\mathbb{R}^n,$ $x\mapsto N_\Gamma(x),$ under the condition that $M_q(x):=q(x)-\Theta$ is metrically subregular at the reference point. Then, based on this formula, we exhibit formulas for computing the graphical derivative of solution mappings and present characterizations of the isolated calmness for a broad class of generalized equations. Finally, applying this to optimization, we get a new result on the isolated calmness of stationary point mappings.