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For a Lipschitz function $f: X \rightarrow X$, Clarke's generalized Jacobian at $x$ is defined as the convex hull of the following set:

$$\delta f (x) = \text{convex hull} \left \{\lim_{x_i \rightarrow x} Jf(x_i) : x_i \rightarrow x, f \text{ is differentiable at } x_i \right\}.$$

Clarke's inverse function theorem says that if every matrix in $\delta f(x)$ is invertible, then there is a Lipschitz inverse function $f^{-1}$ on some neighborhood of $x$.

It seems natural to guess that the generalized Jacobian of the inverse function $f^{-1}$ is just the convex hull of the inverses of all the matrices in the generalized Jacobian of $f$. That is,

$$\delta (f^{-1}) (x) = \text{convex hull} \{M^{-1} : M \in \delta f \}.$$

Is this true? If not, are there some special cases where it holds? I can't find it in Clarke's papers or anywhere else.

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It is true. It is easy to get a formula for Crlarke directional derivative and it should follow from there.

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  • $\begingroup$ Could you add details and explain how do you prove it? $\endgroup$
    – Amir Sagiv
    Jun 12 '20 at 16:22

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