Practical Aspects of the Moreau--Yosida Regularization: Theoretical Preliminaries

Authors:
Claude Lemaréchal;Claudia Sagastizábal
Affiliations:
-;-
Venue:
SIAM Journal on Optimization
Year:
1997

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Abstract

When computing the infimal convolution of a convex function f with the squared norm, the so-called Moreau--Yosida regularization of f is obtained. Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. The most important part of our study concerns second-order differentiability: existence of a second-order development of $f$ implies that its regularization has a Hessian. For the converse, we disclose the importance of the decomposition of ${\Bbb R}^N$ along $\cal U$ (the subspace where f is "smooth") and $\cal V$ (the subspace parallel to the subdifferential of f).