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According to theorem 8.14 of https://arxiv.org/pdf/1708.04180.pdf, we have that for locally Lipschitz $g:Y\to \mathbb R$ and Frechet-differentiable $f: X \to Y$, that

$$(g\circ f)^\circ(x;v) \leq g^\circ(f(x);f'(x)v).$$

Equality holds if $g$ is regular.

According to theorem 8.14 of https://arxiv.org/pdf/1708.04180.pdf, we have that for locally Lipschitz $g:Y\to \mathbb R$ and Frechet-differentiable $f: X \to Y$, that

$$(g\circ f)^\circ(x;v) \leq g^\circ(f(x);f'(x)v).$$

Equality holds if $g$ is regular.

We can furthermore say that:

$$(g\circ f)^\circ(x;v) \geq -g^\circ(f(x);-f'(x)v)$$ under the same conditions.

According to theorem 8.14 of https://arxiv.org/pdf/1708.04180.pdf, we have that for locally Lipschitz $g$ and Frechet-differentiable $f$, that

$$(g\circ f)^\circ(x;v) \leq g^\circ(f(x);f'(x)v).$$

Equality holds if $g$ is regular.

According to theorem 8.14 of https://arxiv.org/pdf/1708.04180.pdf, we have that for locally Lipschitz $g:Y\to \mathbb R$ and Frechet-differentiable $f: X \to Y$, that

$$(g\circ f)^\circ(x;v) \leq g^\circ(f(x);f'(x)v).$$

Equality holds if $g$ is regular.

By lemma 8.8 from here: https://arxiv.org/pdf/1708.04180.pdf, we have that $f^\circ(x;v) = \sup_{v^* \in \partial_C f(x)}\langle v^*,v\rangle$. We must also have that $-f^\circ(x;-v)=\inf_{v^* \in \partial_C f(x)}\langle v^*,v\rangle$. Let $g$ be Frechet differentiable. The chain rule for Clarke's gradient states that $\partial_C (f\circ g)(x) \subseteq g'(x)^* \partial_C f(g(x))$. Taking an inner product with some $v$ gives $(f \circ g)^\circ(x;v)\in[-f^\circ(g(x);-g'(x)v),f^\circ(g(x);g'(x)v)]$, which is a weak chain rule.

According to theorem 8.14 of https://arxiv.org/pdf/1708.04180.pdf, we have that for locally Lipschitz $g$ and Frechet-differentiable $f$, that

$$(g\circ f)^\circ(x;v) \leq g^\circ(f(x);f'(x)v).$$

Equality holds if $g$ is regular.