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

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

If $k=0$, then by definition $g_{0}$ is constant and $g_{0}\equiv \inf \limits _{y\in X}g(y)$.

Examples

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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)