Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 62 / 4 / b / Solution

LetThe proximal operator optimality condition saysBy subgradient inversion under convex conjugacy, which uses Fenchel–Young inequality and $f^{\ast \ast }=f$ from the Fenchel-Moreau theorem,Consequently $z-p=\tau v\in \tau \partial f^{\ast }(p)$, meaning $p=B_{\tau f^{\ast }}(z)$. Uniqueness of the backward subgradient step identifies the result:This is the scaled Moreau decomposition. It requires one backward step on $f$, with reciprocal parameter $1/\tau$ and scaled input $z/\tau$, followed by a scalar multiplication and subtraction. No separate proximal computation of the convex conjugate is needed.