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Rodrigo de Azevedo
Rodrigo de Azevedo
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Reference Request: Differentiability of the Moreau Envelopeenvelope

corrected typo mentioned in the comments and in one answer.
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edit approved Jun 29, 2023 at 17:48
ViktorStein
edit approved Jun 29, 2023 at 17:48
ViktorStein
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I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$ e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z) , $$\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?

I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$ e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z) , $$ 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?

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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ABIM
ABIM
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Reference Request: Differentiability of Moreau Envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$ e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z) , $$ 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?