Moreau envelope

The Moreau envelope (or the Moreau-Yosida regularization) $M_{f}$ of a proper lower semi-continuous convex function $f$ is a smoothed version of $f$. It was proposed by Jean-Jacques Moreau in 1965.

The Moreau envelope has important applications in mathematical optimization: minimizing over $M_{f}$ and minimizing over $f$ are equivalent problems in the sense that the sets of minimizers of $f$ and $M_{f}$ are the same. However, first-order optimization algorithms can be directly applied to $M_{f}$, since $f$ may be non-differentiable while $M_{f}$ is always continuously differentiable. Indeed, many proximal gradient methods can be interpreted as a gradient descent method over $M_{f}$.

Definition

The Moreau envelope of a proper lower semi-continuous convex function $f$ from a Hilbert space ${\mathcal {X}}$ to $(-\infty ,+\infty ]$ is defined as

$M_{\lambda f}(v)=\inf _{x\in {\mathcal {X}}}\left(f(x)+{\frac {1}{2\lambda }}\|x-v\|_{2}^{2}\right).$

Given a parameter $\lambda \in \mathbb {R}$, the Moreau envelope of $\lambda f$ is also called as the Moreau envelope of $f$ with parameter $\lambda$.

Properties

• The Moreau envelope can also be seen as the infimal convolution between $f$ and $(1/2)\|\cdot \|_{2}^{2}$.
• The proximal operator of a function is related to the gradient of the Moreau envelope by the following identity:

$\nabla M_{\lambda f}(x)={\frac {1}{\lambda }}(x-\mathrm {prox} _{\lambda f}(x))$. By defining the sequence $x_{k+1}=\mathrm {prox} _{\lambda f}(x_{k})$ and using the above identity, we can interpret the proximal operator as a gradient descent algorithm over the Moreau envelope.

• Using Fenchel's duality theorem, one can derive the following dual formulation of the Moreau envelope:

$$M_{\lambda f}(v)=\max _{p\in {\mathcal {X}}}\left(\langle p,v\rangle -{\frac {\lambda }{2}}\|p\|^{2}-f^{*}(p)\right),$$

where $f^{*}$ denotes the convex conjugate of $f$. Since the subdifferential of a proper, convex, lower semicontinuous function on a Hilbert space is inverse to the subdifferential of its convex conjugate, we can conclude that if $p_{0}\in {\mathcal {X}}$ is the maximizer of the above expression, then $x_{0}:=v-\lambda p_{0}$ is the minimizer in the primal formulation and vice versa.

• By Hopf–Lax formula, the Moreau envelope is a viscosity solution to a Hamilton–Jacobi equation. Stanley Osher and co-authors used this property and Cole–Hopf transformation to derive an algorithm to compute approximations to the proximal operator of a function.

See also

• Proximal operator
• Proximal gradient method

External links

• "Moreau envelope function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• A Hamilton–Jacobi-based Proximal Operator: a YouTube video explaining an algorithm to approximate the proximal operator