The Moreau-Yosida Regularization

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Motivation

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Definitions

Let $(X,d)$ be a metric space. A function $g:X\to (-\infty ,+\infty ]$ is said to be proper if it is not identically equal to $+\infty$, that is, if there exists $x\in X$ such that $g(x)<+\infty$.

For a given function $g:X\to (-\infty ,+\infty ]$ and $k\geq 0$, its Moreau-Yosida regularization $g_{k}:X\to [-\infty ,+\infty ]$ is given by

$g_{k}(x):=\inf \limits _{y\in X}\left[g(y)+kd(x,y)\right].$

Examples

• If $k=0$, then by definition $g_{0}$ is constant and $g_{0}\equiv \inf \limits _{y\in X}g(y)$.
• If $g$ is not proper, then $g_{k}=+\infty$ for all $k\geq 0$.

Take $(X,d):=(\mathbb {R} ,|\cdot |)$. If $g$ is finite-valued and differentiable, we can explicitly write down $g_{k}$. Then for a fixed $x\in \mathbb {R}$, the map $g_{k,x}:y\mapsto y^{2}+k|x-y|$ is continuous everywhere and differentiable everywhere except for when $y=x$, where the derivative does not exist due to the absolute value. Thus we can apply standard optimization techniques from Calculus to solve for $g_{k}(x)$: find the critical points of $g_{k,x}$ and take the infimum of $g_{k,x}$ evaluated at the critical points. One of these values will always be the original function $g$ evaluated at $x$, since this corresponds to the critical point $y=x$ for $g_{k,x}$.

• Let $g(x):=x^{2}$. Then

$g_{k}(x)=\min \left\{x^{2},{\frac {k^{2}}{2}}+k\left|x\pm {\frac {k}{2}}\right|\right\}.$

Results

Proposition.

• If $g$ is proper and bounded below, so is $g_{k}$. Furthermore, $g_{k}$ is continuous for all $k\geq 0$.
• If, in addition, $g$ is lower semicontinuous, then $g_{k}(x)\nearrow g(x)$ for all $x\in X$.
• In this case, $g_{k}\wedge k:=\min(g_{k},k)$ is continuous and bounded and $g_{k}(x)\wedge k\nearrow g(x)$ for all $x\in X$.

References

Possible list of references, will fix accordingly

Bauschke-Combette Ch 12.; Santambrogio (6); Ambrosio-Gigli-Savare (59-61)