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
Famous quotes containing the words mathematical and/or theorem:
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)