Compensated convexity and its applications
Kewei Zhang · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2007
We introduce the notions of lower and upper quadratic compensated convex transforms C_{2,\lambda }^{l}(f) and C_{2,\lambda }^{u}(f) respectively and the mixed transforms by composition of these transforms for a given function f\:\text{:}\mathbb{R}^{n}\mapsto \mathbb{R} and for possibly large \lambda > 0 . We study general properties of such transforms, including the so-called ‘tight’ approximation of C_{2,\lambda }^{l}(f) to f as \lambda \rightarrow + \infty 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,\lambda }^{l}(\mathrm{dist}^{2}(x,K)) of the squared-distance function to a compact set K and the quadratic upper transform C_{2,\lambda }^{u}(f) for any convex function f of at most quadratic growth. We show that both C_{2,\lambda }^{l}(\mathrm{dist}^{2}(x,K)) and C_{2,\lambda }^{u}(f) are C^{1,1} approximations of the original functions for large \lambda > 0 and C_{2,\lambda }^{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.