10.10. Conjugate Duality#
10.10.1. Fenchel’s Duality Theorem#
Consider the minimization problem
The problem can be rewritten as
Construct the Lagrangian for this problem.
The dual objective is constructed by minimizing the Lagrangian with the primal variables \(\bx, \bz\).
We thus obtain the following dual problem, known as the Fenchel’s dual:
(Fenchel’s duality theorem)
Let \(f,g : \VV \to \RERL\) be proper convex functions. If \(\relint \dom f \cap \relint \dom g \neq \EmptySet\), then
The supremum of R.H.S. (the dual problem) is attained whenever it is finite.