What is a Fenchel conjugate?

The Fenchel conjugate is a conjugate of a function. It is the generalization of the self-inverse transformation of real-valued convex functions. It also applies to non-convex functions.

The Fenchel conjugate

Let's say there is a vector space$X$with its dual$X^*$(containing the linear functions of$X$). Then,

is its canonical dual pairing where $x \in X$. If there is a function

then

is its convex conjugate. If $x^* \in X^*$ is the supremum, which means that if $X^*$ is the subset of, let's say, a vector space, $P$, then $x^*$ is the most minor element in $P$ that is greater or equal to each element present in $X^*$. In such a scenario, the conjugate becomes:

An equivalent conjugate to this would be something at the infimum, which is when $x^*$ is the greatest element in $P$ that is less than or equal to each element in $X^*$. In this case, the conjugate can be written as:

Properties

Some of the properties of Convex conjugates are listed below.

Reverse order

If,

then it follows that:

Lower semi-continuous biconjugate

The convex conjugate of a function is always lower semi-continuous, meaning that there is a point at which all other nearby points have a function value less than or equal to that point. The bi-conjugate is the largest lower semi-continuous function.

Convexity

If there are two functions,$f$ and $g$, then for $\lambda$ residing between $0$ and $1$, the convexity relation becomes:

Linear Transformations

A closed convex function $f$ is symmetric with respect to an orthonormal matrixMatrix with each vector having a magnitude of 1 and these vectors being orthogonal to each other. $G$

where,

only if its conjugate $f^*$ is symmetric with respect to $G$.

Example

Convex conjugate of power function

The convex conjugate of a power function

can be written as,

Convex conjugate of an exponential equation

The convex conjugate of an exponential equation

can be written as,