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.

Fenchel-Moreau Theorem

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

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

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

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

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

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

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

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

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

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

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 proper, lower semicontinuous, convex function $f:X\to \mathbb {R} \cup \{+\infty \}$, the primal optimization problem is given by

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

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 $F:X\times U\to \mathbb {R} \cup \{+\infty \}$ be a proper convex function so that $f(x)=F(x,0)$. Then the corresponding primal and dual problems may be written as

$P_{0}:=\inf _{x\in X}F(x,0)$

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

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

$P_{0}=P(0)$

$D_{0}=P^{**}(0)$

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.