A generalization of the Moreau-Yosida regularization

Aras Bacho

Published 2022-01-11Version 1

For a function $f:X \rightarrow (-\infty,+\infty]$ on a normed space $(X,\Vert \cdot \Vert)$ and given parameters $p>1$ and $\varepsilon>0$, we investigate the properties of the generalized Moreau-Yosida regularization given by \begin{align*} f_\varepsilon(u)=\inf_{v\in X}\left\lbrace \frac{1}{p\varepsilon} \Vert u-v\Vert^p+f(v)\right\rbrace \quad ,u\in X. \end{align*} We show that the generalized Moreau-Yosida regularization satisfies the same properties as in the classical case for $p=2$ provided $X$ is not a Hilbert space.