Moreau envelope ($M_{t}f$)

The Moreau envelope of a proper closed convex function isIt is differentiable even when $f$ is not, with $\nabla M_{t}f(x)=[x-{\mathrm{prox}}_{tf}(x)]/t$ and $1/t$-Lipschitz gradient.

For $r=\Vert x{\Vert }_2$, the Moreau envelope of the Euclidean norm is $r^2/(2\tau )$ when $r\le \tau$ and $r-\tau /2$ otherwise. Its gradient is $x/\tau$ in the core and $x/r$ outside. The approximation error is between zero and $\tau /2$.

For proper lower semicontinuous convex functions, the Moreau envelope has gradient $\nabla f_{\tau }=(I-{\mathrm{prox}}_{\tau f})/\tau$. 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.

For a proper convex function with sequential lower semicontinuity $U=-\log \mu$, form $U_{\lambda }(x)={\inf }_y\{U(y)+\lambda \Vert x-y{\Vert }^2\}$ with $\lambda >0$. The Moreau envelope theorem gives $\nabla U_{\lambda }(x)=2\lambda (x-{\mathrm{prox}}_{U/(2\lambda )}(x))$. If $e^{-U_{\lambda }}$ is integrable, it defines a smooth surrogate probability density function. Infimizing the logarithm of a probability density function with a positive quadratic penalty has the opposite sign and does not generally smooth it. For the Laplace distribution, that incorrect operation leaves $-\log 2-|x|-1/(4\lambda )$, which is not differentiable at zero.

For a nonempty closed convex set $C$, the squared-distance function is the Moreau envelope of its indicator functional of a constraint set. It is convex and differentiable withand this gradient is one-Lipschitz.

The squared distance to a convex set is the infimal convolution of the set's indicator functional of a constraint set and the half-squared Hilbert space norm. Its convex conjugate is the sum of the half-squared norm and the support function of the set.