  In this paper we study how Lagrange duality is connected to optimization
problems whose objective function is the difference of two convex functions,
briefly called DC problems. We present two Lagrange dual problems, each of them
obtained via a different approach. While one of the duals corresponds to the
standard formulation of the Lagrange dual problem, the other is written in
terms of conjugate functions. When one of the involved functions in the
objective is evenly convex, both problems are equivalent, but this relation is
no longer true in the general setting. For this reason, we study conditions
ensuring not only weak, but also zero duality and strong duality between the
primal and one of the dual problems written using conjugate functions. For the
other dual, and due to the fact that weak duality holds by construction, we
just develop conditions for zero duality gap and strong duality between the
primal DC problem and its (standard) Lagrange dual problem. Finally, we
characterize weak and strong duality together with zero duality gap between the
primal problem and its Fenchel-Lagrange dual following techniques used
throughout the manuscript.
