2.3 Convex Conjugates

Convex Conjugate

The convex (Fenchel) conjugate $f^{\ast }:X^{\ast }\to \mathbb{R}$ of a function $f:X\to \mathbb{R}$ is used to represent $f$ as a set of supporting hyperplanes with slopes $x^{\ast }$, such that the parameters of each $x^{\ast }$ are encoded in $f^{\ast }$. Then, $f^{\ast }(x^{\ast })$ is the intercept at which the line $⟨x^{\ast },x⟩$ becomes a tangent of $f$. Formally,
$f^{\ast }(x^{\ast })={\sup }_{x\in X}\{⟨x^{\ast },x⟩-f(x)\},$
such that $t(x)=⟨x^{\ast },x⟩-f^{\ast }(x^{\ast })$.

Preservation of Convexity

The convex conjugate of any function is always a convex function.

Properties

(Fenchel’s Inequality) Any conjugate pair $f,f^{\ast }$ satisfies $⟨x^{\ast },x⟩\le f(x)+f^{\ast }(x^{\ast })$.

(Weak Duality of Biconjugation) For any $f$, it follows that $f^{\ast \ast }\le f$, such that $f^{\ast \ast }=f$ if and only if $f$ is convex and l.s.c.

(Affine Conjugate) Any $a(x)=⟨x_a^{\ast },x⟩+b$ has $a^{\ast }(x^{\ast })=\{\begin{array}{ll}-b& x^{\ast }=x_a^{\ast }\\ +\infty & x^{\ast }\ne x_a^{\ast }\end{array}$, and $a^{\ast \ast }(x)=a(x)$.

Examples

Economic Interpretation

Let $x$ be the quantity of some goods you are trying to sell (i.e., laptops, tablets, phones, etc.). Then $f:X\to \mathbb{R}$ can be defined as the total cost of producing the goods (i.e., the dot product between $x$ and the respective cost of producing 1 unit of each good). Take the Fenchel conjugate of $f(x)$:
$\underset{\text{Profit}}{\underbrace{f^{\ast }(x^{\ast })}}={\sup }_{x\in X}\{\underset{\text{Revenue}}{\underbrace{⟨x^{\ast },x⟩}}-\underset{\text{Cost}}{\underbrace{f(x)}}\}.$
Then, $f^{\ast }:X^{\ast }\to \mathbb{R}$ represents the optimal profit given the goods are sold at some prices $x^{\ast }$ per 1 unit of goods.