Compensated convexity and its applications

We introduce the notions of lower and upper quadratic compensated convex transforms and respectively and the mixed transforms by composition of these transforms for a given function and for possibly large λ>0. We study general properties of such transforms, including the so-called ‘tight’ approximation of 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 of the squared-distance function to a compact set K and the quadratic upper transform for any convex function f of at most quadratic growth. We show that both and are C1,1 approximations of the original functions for large λ>0 and 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.