# Fenchel-Moreau and Primal/Dual Optimization Problems

The Fenchel-Moreau Theorem[1] 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.[2]

## Fenchel-Moreau Theorem

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

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

## 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 ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$, its convex conjugate ${\displaystyle f^{*}:X^{*}\to \mathbb {R} \cup \{+\infty \}}$ is defined by

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

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

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

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

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

As above, the biconjugate function f** is always convex and lower semicontinuous. Furthermore, by a second application of Young's inequality, we have

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

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 ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$, the primal optimization problem is given by

${\displaystyle \inf _{x\in X}f(x).}$

The corresponding dual problem arises from a suitable perturbation" of the primal problem, subject to a parameter uU, where U is also a normed vector space. In particular, let ${\displaystyle F:X\times U\to \mathbb {R} \cup \{+\infty \}}$ be a proper convex function so that ${\displaystyle f(x)=F(x,0)}$. Then the corresponding primal and dual problems may be written as

${\displaystyle P_{0}:=\inf _{x\in X}F(x,0)}$
${\displaystyle D_{0}:=\sup _{v\in U^{*}}-F^{*}(0,v)}$

The formulation of these problems becomes even simpler from the perspective of the inf-projection ${\displaystyle P(u):=\inf _{x}F(x,u)}$. With this notation, the primal and dual problems are given by

${\displaystyle P_{0}=P(0)}$
${\displaystyle D_{0}=P^{**}(0)}$

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.