Abstract
AbstractClarke’s inverse function theorem for Lipschitz mappings states that a bi-Lipschitz mapping f is locally invertible about a point $$x_0$$
x
0
if the generalized Jacobian $$\partial f(x_0)$$
∂
f
(
x
0
)
does not contain singular matrices. It is shown that under these assumptions the generalized Jacobian of the inverse mapping at $$f(x_0)$$
f
(
x
0
)
is the convex hull of the set of matrices that can be obtained as limits of sequences $$J_f(x_k)^{-1}$$
J
f
(
x
k
)
-
1
with f differentiable in $$x_k$$
x
k
and $$x_k$$
x
k
converging to $$x_0$$
x
0
. This identity holds as well if f is assumed to be locally bi-Lipschitz at $$x_0$$
x
0
.
Funder
Deutsche Forschungsgemeinschaft
Universität Regensburg
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Management Science and Operations Research,Control and Optimization
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