Journal of Convex Analysis 15 (2008), No. 1, 179--190
Copyright Heldermann Verlag 2008
Self-Dual Smoothing of Convex and Saddle Functions
Rafal Goebel
3518 NE 42 Street, Seattle, WA 98105, U.S.A.
rafal.k.goebel@gmail.com
It is shown that any convex function can be approximated by a family of differentiable with Lipschitz continuous gradient and strongly convex approximates in a "self-dual" way: the conjugate of each approximate is the approximate of the conjugate of the original function. The approximation technique extends to saddle functions, and is self-dual with respect to saddle function conjugacy and also partial conjugacy that relates saddle functions to convex functions.
Keywords: Convex functions, approximation, Moreau envelopes, duality, saddle functions.
MSC: 52A41, 90C25, 90C59, 90C46, 26B25
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