Compensated convexity and its applications

Abstract

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 $\lambda \to +\infty$ 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.