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

We consider the optimization problem (PA) infx∈ X(x)+g(Ax)\ where f and g are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y, respectively, and A is a linear operator from X to Y. By using the properties of the epigraph of the conjugated functions, some sufficient and necessary conditions for the strong Fenchel duality and the strong converse Fenchel duality of (PA) are provided. Sufficient and necessary conditions for the stable Fenchel duality and for the total Fenchel duality are also derived.