A generalization of the Moreau-Yosida regularization

TL;DR

This paper extends the classical Moreau-Yosida regularization by replacing the quadratic kernel with a more general function on normed spaces, analyzing its properties and convergence behavior.

Contribution

It introduces a generalized form of Moreau-Yosida regularization applicable to broader spaces and establishes key properties and convergence results.

Findings

Generalized regularization retains classical properties in non-Hilbert spaces.

Proves Mosco-convergence as regularization parameter approaches zero.

Extends regularization techniques to a wider class of functions and spaces.

Abstract

In many applications, one deals with nonsmooth functions, e.g., in nonsmooth dynamical systems, nonsmooth mechanics, or nonsmooth optimization. In order to establish theoretical results, it is often beneficial to regularize the nonsmooth functions in an intermediate step. In this work, we investigate the properties of a generalization of the Moreau-Yosida regularization on a normed space where we replace the quadratic kernel in the infimal convolution with a more general function. More precisely, for a function $f:X \rightarrow (-\infty,+\infty]$ defined 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 that $X$ is not a Hilbert space. We further establish a convergence result in the sense of Mosco-convergence as the regularization parameter $\varepsilon$ tends to zero.