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Computation of Graphical Derivative for a Class of Normal Cone Mappings under a Very Weak Condition

Huy Chieu Nguyen(nghuychieu***at***gmail.com)
Van Hien Le(lehiendhv***at***gmail.com)

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

Keywords: Computation, Graphical Derivative, Normal Cone Mapping, Generalized Equation, Isolated Calmness

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

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Entry Submitted: 03/29/2016
Entry Accepted: 03/29/2016
Entry Last Modified: 03/29/2016

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