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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g : X \to (-\infty,+\infty]} and $k\geq 0$, its Moreau-Yosida regularization Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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References

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