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 the equivalence of primal and dual optimization problems.

Theorem Statement

Given a normed vector space X and $f:X\to \mathbb {R} \cup \{+\infty \}$, then

$f{\text{ if convex and lower semicontinuous}}\iff f^{**}=f.$

Background on Convex Conjugate Functions

LetX be a normed vector space, and let X* denote its topological dual. Given an extended real-valued function $f:X\to \mathbb {R} \cup \{+\infty \}$, its convex conjugate $f^{*}:X^{*}\to \mathbb {R} \cup \{+\infty \}$ is defined by

$f^{*}(u):=\sup _{x\in X}\{\langle u,x\rangle -f(x)\}\quad \forall u\in X^{*}.$

An immediate consequence of this definition is Young's Inequality,

$f^{*}(u)+f(x)\geq \langle u,x\rangle \quad \forall x\in X,u\in X^{*}.$

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 $f^{**}:X\to \mathbb {R} \cup \{+\infty \}$ is defined by

$f^{**}(x):=\sup _{u\in X^{*}}\{\langle u,x\rangle -f^{*}(u)\}\quad \forall u\in X^{*}.$

As above, for any function f, its biconjugate function f^* is convex and lower semicontinuous. Furthermore, by Young's inequality, we always have

$f^{**}(x)\leq f(x)\quad \forall x\in X.$

In this way, it is clear that, in order for equality to hold, it is necessary for f to be convex and lower semicontinuous. The heart of Fenchel-Moreau Theorem is that this is also sufficient.