论文标题
Moreau-Yosida正则化的概括
A generalization of the Moreau-Yosida regularization
论文作者
论文摘要
在许多应用程序中，一个人涉及非平滑功能，例如，在非平滑动力学系统，非平滑力学或非滑动优化中。为了建立理论结果，在中间步骤中规范非平滑函数通常是有益的。在这项工作中，我们研究了Moreau-Yosida正则化在规范空间上的概括的特性，在该空间中，我们用更通用的功能代替了虚拟卷积中的二次内核。更准确地说，对于功能$ f：x \ rightarrow（ - \ infty，+\ \ infty] $在规范空间$（x，x，x，\ vert \ cdot \ vert）$和给定参数$ p> 1 $和$ \ \ varepsilon> 0 $中f_ \ varepsilon（u）= \ inf_ {v \ in x} \ left \ lbrace \ lbrace \ frac {1} {p \ varepsilon} \ varepsilon} \ vert u-v \ vert u-v \ vert^p+f（v）如果$ x $不是Hilbert空间，则与经典案例相同，我们进一步建立了Mosco-Convergence的融合结果。
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.
