Characterization and recognition of d.c. functions

A function $${f : \Omega \to \mathbb{R}}$$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g − h with $${g, h : \Omega \to \mathbb{R}}$$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C1,1 functions and lower-C2 functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function $${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry.