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
- f and g are lower semi-continuous and where is the algebraic interior and where h is some function is the set, or
- where are the points where the function is continuous.
Then strong duality holds, i.e. . If then supremum is attained.
Read more about this topic: Fenchel's Duality Theorem
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