Fenchel's Duality Theorem - Mathematical Theorem

Mathematical Theorem

Let X and Y be Banach spaces, and be convex functions and be a bounded linear map. Then the Fenchel problems:

satisfy weak duality, i.e. . Note that are the convex conjugates of f,g respectively, and is the adjoint operator. The perturbation function for this dual problem is given by .

Suppose that f,g, and A satisfy either

1. f and g are lower semi-continuous and where is the algebraic interior and where h is some function is the set, or
2. where are the points where the function is continuous.

Then strong duality holds, i.e. . If then supremum is attained.