I've recently come across many results discussing the differentiation of the Moreau envelope defined by
\begin{equation}
e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(h),
\end{equation}
where $f$ is a convex functional on a separable Hilbert space $H$.
Examples of results on differentiability are here, as well as Moreau's original papers. However, I didn't seem to come across any results about the twice differentiability of the operator... Are these results known/ what are the conditions on a convex function $f$ so that its Moreau envelope is twice continuously differentiable?
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2 Answers
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This is not an answer, but a few thoughts on the subject:
As you know, the Moreau enveloppe of a convex lower semicontinuous function is always Fréchet differentiable, with its gradient being $1$-lipschitz continuous. Actually, its gradient can be expressed as follows: $\nabla e(f)(x) = x - \text{prox}_f (x)$, where $\text{prox}_f$ is the proximal operator of $f$. So questioning the differentiability of $\nabla e(f)$ is equivalent to the one of $prox_f$.
One particular interesting example is $f$ being the indicator function of a closed convex set $C$ with nonempty interior, for which $prox_f$ becomes the projection $\text{proj}_C$. In this case, we know 1 that the projection is differentiable (almost) if and only if the set has a $C^2$ boundary.
For general functions $f$, we have that $x^+ = \text{prox}_f(x) \iff x^+ = \text{proj}_{S_f(x^+)}(x)$, where $S_f(x^+)$ is the sublevel set $\{y \in H \ | \ f(y) \le f(x^+) \}$. Even though $S_f(x^+)$ depends on $x$, this suggests that the projection onto sublevel sets should be differentiable. According to what precedes, we infer informally that the function should have lower sets of class $C^2$ (e.g. $f$ having a $C^2$ domain, and being twice differentiable in this domain). I unfortunately found no results on this question.
1 J.-B. Hiriart-Urruty. “At What Points Is the Projection Mapping Differentiable?” The American Mathematical Monthly, vol. 89, no. 7, 1982, pp. 456–458.
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Terry Rockafellar has a paper on this topic in SIAM Optimization titled Generalized Hessian Properties of Regularized Nonsmooth Functions. See theorem 3.8. Rockafellar proves an equivalence between twice-differentiability of the Moreau envelope and twice epi-differentiability of the function $f$.
