Gradient of a Moreau envelope
= Gradient of a Moreau envelope
{title2=$\nabla f_\tau=(I-\operatorname{prox}_{\tau f})/\tau$}
For proper <lower semicontinuous> <convex functions>, the <Moreau envelope> has gradient $\nabla f_\tau=(I-\operatorname{prox}_{\tau f})/\tau$. This follows by <subgradient inversion under convex conjugacy> and the quadratic conjugate. Firm nonexpansiveness of the proximal residual makes the gradient $1/\tau$-Lipschitz continuous.