Fenchel-Lagrange Duality for Dc Programs with Composite Functions

In this paper, we present some duality results for a DC programming problem (P) involving a composite function. To this end, by using the standard convexification technique, we first introduce a Fenchel-Lagrange dual problem for (P). Then, under a closedness qualification condition, we prove the existence of the strong duality for (P). The strong duality result is then applied to obtain an extended Farkas lemma and an alternative type theorem for (P). Moreover, as applications of these results, a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator are examined at the end of this paper.