Fenchel-Young inequality

In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.

Definition

Let $X$ be a real topological vector space and let $X^{*}$ be the dual space to $X$ . Denote by

\langle \cdot ,\cdot \rangle :X^{*}\times X\to \mathbb {R}

the canonical dual pairing, which is defined by $\left(x^{*},x\right)\mapsto x^{*}(x).$

For a function $f:X\to \mathbb {R} \cup \{-\infty ,+\infty \}$ taking values on the extended real number line, its convex conjugate is the function

f^{*}:X^{*}\to \mathbb {R} \cup \{-\infty ,+\infty \}

whose value at $x^{*}\in X^{*}$ is defined to be the supremum:

f^{*}\left(x^{*}\right):=\sup \left\{\left\langle x^{*},x\right\rangle -f(x)~\colon ~x\in X\right\},

or, equivalently, in terms of the infimum:

f^{*}\left(x^{*}\right):=-\inf \left\{f(x)-\left\langle x^{*},x\right\rangle ~\colon ~x\in X\right\}.

This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.^[1]

Examples

For more examples, see § Table of selected convex conjugates.

[1]