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- W2014718373 abstract "Abstract We introduce the notions of lower and upper quadratic compensated convex transforms C 2 , λ l ( f ) and C 2 , λ u ( f ) respectively and the mixed transforms by composition of these transforms for a given function f : R n ↦ R and for possibly large λ > 0 . We study general properties of such transforms, including the so-called ‘tight’ approximation of C 2 , λ l ( f ) to f as λ → + ∞ and compare our transforms with the well-known Moreau–Yosida regularization (Moreau envelope) and the Lasry–Lions regularization. We also study analytic and geometric properties for both the quadratic lower transform C 2 , λ l ( dist 2 ( x , K ) ) of the squared-distance function to a compact set K and the quadratic upper transform C 2 , λ u ( f ) for any convex function f of at most quadratic growth. We show that both C 2 , λ l ( dist 2 ( x , K ) ) and C 2 , λ u ( f ) are C 1 , 1 approximations of the original functions for large λ > 0 and C 2 , λ u ( f ) remains convex. Explicitly calculated examples of quadratic transforms are given, including the lower transform of squared distance function to a finite set and upper transform for some non-smooth convex functions in mathematical programming." @default.
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- W2014718373 date "2008-08-01" @default.
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- W2014718373 title "Compensated convexity and its applications" @default.
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- W2014718373 doi "https://doi.org/10.1016/j.anihpc.2007.08.001" @default.
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