Computation of graphical derivatives of normal cone maps to a class of conic constraint sets

TL;DR

This paper derives an exact formula for the graphical derivative of normal cone maps to a broad class of conic constraints, advancing the understanding of solution stability in nonconvex optimization problems.

Contribution

It provides a novel characterization of the graphical derivative for normals to nonconvex conic constraints without assuming nondegeneracy.

Findings

Exact characterization of the graphical derivative for nonconvex conic constraints

Applicable to conic sets that are $C^2$-cone reducible

No nondegeneracy condition required for the main results

Abstract

This paper concerns with the graphical derivative of the normals to the conic constraint $g(x)\in\!K$, where $g\!:\mathbb{X}\to\mathbb{Y}$ is a twice continuously differentiable mapping and $K\subseteq\mathbb{Y}$ is a nonempty closed convex set assumed to be $C^2$-cone reducible. Such a generalized derivative plays a crucial role in characterizing isolated calmness of the solution maps to generalized equations whose multivalued parts are modeled via the normals to the nonconvex set $\Gamma=g^{-1}(K)$. The main contribution of this paper is to provide an exact characterization for the graphical derivative of the normals to this class of nonconvex conic constraints under an assumption without requiring the nondegeneracy of the reference point as the papers \cite{Gfrerer17,Mordu15,Mordu151} do.