Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces

We consider the DC optimization problem (P)infx∈X(f1(x)-f2(x))+(g1(Ax)-g2(Ax)), wheref1,f2,g1andg2are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y respectively, and A is a linear continuous operator from X to Y. Adopting the standard convexification technique, a Fenchel dual problem of (P) is given. By using properties of the epigraph of conjugate functions, some sufficient and necessary conditions for the Fenchel duality and for the stable Fenchel duality of (P) are provided.