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))}, where f1,f2,g1f1,f2,g1 and g2g2 are proper convex functions defined on locally convex Hausdorff topological vector spaces XX and YY respectively, and AA is a linear continuous operator from XX to YY. Adopting the standard convexification technique, a Fenchel dual problem of (P)(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)(P) are provided.