On proto-differentiability of generalized perturbation maps

Authors: Gue Myung Lee, Nguyễn Quang Huy,

https://doi.org/10.1016/j.jmaa.2006.01.030

Publisher, magazine: ,

Publication year: 2006

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Abstract

Main result: Under some suitable qualification conditions, the generalized perturbation map \(G\) (that is, the solution set map to a parameterized constraint system, to a parameterized variational inequality, or to a parameterized optimization problem) in \(G(x,z)=\{ y\in D: z\in F(x,y)+K(y)\}\) (where \(F:R^d\times R^n = R^m , K: R^n = R^m)\) is proto-differentiable. The precise proofs of the main results are proposed.

Tags: optimization; variational inequality; Fréchet differentiable function; closed convex subset in \(R^n\); contingent cone; adjacent cone; multifunction; perturbation map; proto-differentiability