Abstract
We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 ∈f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz ∈ g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)−1 has a local selection which is Lipschitz continuous nearx 0 and Fréchet (Gateaux, Bouligand, directionally) differentiable atx 0 if and only if the linearization inverse (f (x 0) +∇ f (x0) (× − x0) + F(×))−1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.
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This work was supported by National Science Foundation Grant Number DMS 9404431.
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Dontchev, A.L. Implicit function theorems for generalized equations. Mathematical Programming 70, 91–106 (1995). https://doi.org/10.1007/BF01585930
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DOI: https://doi.org/10.1007/BF01585930