Stable and Total Fenchel Duality for DC Optimization Problems in Locally Convex Spaces

We consider the DC (difference of two convex functions) optimization problem ðPÞ inf x∈Xfðf 1ðxÞ− f 2ðxÞÞ þ ðg1ðAxÞ− g2ðAxÞÞg, where f 1, f 2, g1, and g2 are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y , and A is a linear continuous operator from X to Y . Adopting different tactics, two types of the Fenchel dual problems of ðPÞ are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of ðPÞ are provided. Sufficient and/or necessary conditions for the strong Fenchel duality, the stable Fenchel duality, and the stable total duality are derived.