Proximal operator of conjugate function

Question

Asumme $f$ is closed convex function and $f^*$ is the conjugate function. Domain of $f^*$ is $(0,1)$, otherwise $f^*$ is $infty$. If we directly compute  $textrm{prox}_{f^*}(x) $, the result will be in the domain of $f^*$.
From Moreau’s identity we know that
begin{equation}label{1}
textrm{prox}_{f^*}(x) = x - textrm{prox}_{f}(x). 
end{equation}
If we use this equation, is it true that the result is also in the domain of $f^*$?

Zim · Answer

Moreau's identity always holds for proper convex lower-semicontinuous functions (sometimes called closed convex, depending on the book; I'm assuming these are the hypotheses you mean). For any element of a real Hilbert space $xinmathcal{X}$,
begin{equation}
y=textrm{prox}_{f^*}x=(textrm{Id}+partial f^*)^{-1}x;;Leftrightarrow;;xin y+partial f^*y;;Rightarrow partial f^*(y)neqvarnothing,
end{equation}
and since $y=textrm{prox}_{f^*}xin{xinmathcal{X}, |, partial f^*(x)neqvarnothing}subsettextrm{dom} f^*$ you're done.

Proximal operator of conjugate function

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