Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 339 / 2 / b / Solution

For a proper lower-semicontinuous convex function $f$, its proximal operator isThe squared norm is strongly convex, so the minimizer is unique. The subdifferential sum rule gives the necessary and sufficient conditionMore generally,The subgradient inversion rule for the convex conjugate says $u\in \partial f(x)$ exactly when $x\in \partial f^{\ast }(u)$. Hencewhich is precisely the proximal optimality conditionSince $x=y-tu$, this proves the generalized Moreau decomposition