4. DC Functions and DC Sets

A dc function is a function which can be expressed as the difference of two convex functions. A dc set is a set which can be expressed as the difference of two convex sets. Several important properties of dc functions and dc sets are discussed, among them: (1) any continuous function can be approximated as closely as desired by a dc function; (2) any closed set in \(\mathbb{R}^{n}\) is the projection of a dc set from \(\mathbb{R}^{n+1}\); (3) the class of dc functions is stable under linear operations and under finite upper or lower envelope. Due to the latter property, any finite system of dc inequalities can be rewritten equivalently as a single dc inequality. This permits a classification of dc optimization problems into a few basic classes, which is very convenient for their systematic study. The following questions are discussed next: which functions are dc; how to find an effective representation of a dc function, and how to recognize a minimizer of a dc function on \(\mathbb{R}^{n}\). In the last section Toland duality relation in dc minimization problems is presented.