Let $\gamma\in\left]0,+\infty\right[$, let $f$ be a proper, convex, lower semicontinuous function from a real Hilbert space $\mathcal{X}$ to $\left]-\infty,+\infty\right]$, and set $g=f-\frac{\gamma}{2}\|\cdot\|^2$. Then $g$ is weakly convex.
I'm looking for a reference characterizing for which $x\in\mathcal{X}$ the following holds
\begin{equation}
\partial_{Clarke} g(x) = \partial_{convex} f(x) – \gamma x. \tag{*}
\end{equation}
where $\partial_{Clarke}$ is the Clarke subdifferential,
$$\partial_{convex} f(x) = \left\{ u \in \mathcal{X} \mid (\forall y \in \mathcal{X}) \quad \langle y – x \mid u \rangle + f(x)\leq f(y) \right\},$$
and the righthand side in (*) denotes Minkowski subtraction. I know that $(\nabla \frac{\gamma}{2}\|\cdot\|^2) (x) = \gamma x$ and that $\partial_{convex}$ coincides with $\partial_{Clarke}$ on convex functions. However, I am only working with a weakly convex function. I've perused Rockafellar/Wets but not found much.
I'm actually not entirely positive that (*) is true everywhere, e.g. it may fail on the boundary of the domain of $g$. Any relevant info is greatly appreciated!
EDIT: I think it would suffice to find a reference for when the sum rule holds for Clarke subdifferentials. I believe that $\partial_{Clarke}g(x)\supset\partial_{convex}f(x)-\gamma x$, so it would suffice to show the reverse inclusion.
Best Answer
@Zim Optimization and Nonsmooth Analysis Thm 2.9.8, p. 102