Fenchel-Moreau and Primal/Dual Optimization Problems
The Fenchel-Moreau Theorem is a fundamental result in convex analysis, characterizing the class of functions for which a function equals its biconjugate. A key consequence of this theorem is that is provides sufficient conditions for the equivalence of primal and dual optimization problems.
Given a normed vector space X and , then
Background on Convex Conjugate Functions
Let X be a normed vector space, and let X* denote its topological dual. Given an extended real-valued function , its convex conjugate is defined by
An immediate consequence of this definition is Young's Inequality,
Furthermore, it follows directly from the definition that, for any function f, its conjugate function f* is convex and lower semicontinuous.
In a similar way, for any function f, its the biconjugate function is defined by
As above, the biconjugate function f** is always convex and lower semicontinuous. Furthermore, by a second application of Young's inequality, we have
Since f** is always convex and lower semicontinuous, in order for equality to hold for all x, it is necessary that f also be convex and lower semicontinuous. The heart of Fenchel-Moreau Theorem is that this condition is not just necessary, but sufficient.
Application to Primal/Dual Optimization Problems
An important consequence of the Fenchel-Moreau Theorem is that it provides sufficient conditions for the equivalence of primal and dual optimization problems. Given a normed vector space X and a lower semicontinuous, convex function , the primal optimization problem is given by
The corresponding dual problem arises from a suitable ``perturbation" of the primal problem, subject to a parameter u ∈ U, where U is also a normed vector space. In particular, let be a proper convex function so that . Then the corresponding primal and dual problems may be written as
The formulation of these problems becomes even simpler from the perspective of the inf-projection . With this notation, the primal and dual problems are given by
Therefore, by the Fenchel-Moreau theorem, a sufficient condition for equivalence of the primal and dual problems is that the inf-projetion function P(u) is convex and lower semicontinuous.