What does the "conjugate" word in conjugate function mean?

Question

In optimization, we define the conjugate function as: $$f^\star(y) = \sup_{x\in \operatorname{dom}(f)} (y^Tx-f(x))$$

In maths, typically the word conjugate refers to pairs such that $(f^\star)^\star = f$. Does the above conjugate function have any such relation ?

Apologise for the bad formatting.

Answer 1

There's a name for $f^*$: the Legendre transform of $f$. Yes, the property $(f^*)^* = f$ holds. In general, $\ast$ is used to denote an involutive operation (in the sense that it squares to the identity -- other examples being the adjoint operation $T \mapsto T^*$ of a linear operator in a Hilbert space to its adjoint, and actual conjugation in $\Bbb C$).