TITLE: Physical interpretation and notions about conjugate function?
QUESTION [18 upvotes]: If given a convex function $f: \mathbb{R} \to \mathbb{R}$, then the conjugate function $f^*$ is defined as $$f^*(s) = \sup_{t \in \mathbb{R}} (st-f(t))$$
Now i want to understand what is the physical interpretation of this conjugate function? What is its exposition? Please help me.

REPLY [23 votes]: The basic idea behind duality in convex analysis is to view a (closed) convex set $C$ as an intersection of half spaces.  Applying this idea to the epigraph of a convex function $f$ suggests that we should view $f$ as a supremum of affine functions.
An affine minorant of $f$ is a function $x \mapsto \langle m, x \rangle - b$ such that
$$
\tag{$\spadesuit$} f(x) \geq \langle m, x \rangle - b \quad \text{for all } x.
$$
The vector $m$ is called the "slope" of the affine minorant.
Typically $f$ has many affine minorants with a given slope $m$, corresponding to different values of the scalar $b$.  We only care about the best affine minorant with slope $m$ --- in other words, we only care about the best scalar $b$.  So: For a given $m$, which value of $b$ is the "best"?  Which value of $b$ makes the inequality in $(\spadesuit)$ as tight as possible?
Notice that
\begin{align}
& f(x) \geq \langle m, x \rangle - b \quad \text{for all } x \\
\iff & b \geq \langle m, x \rangle - f(x) \quad \text{for all } x\\
\iff & b \geq \sup_x \, \langle m, x \rangle - f(x) = f^*(m).
\end{align}
This shows that the best choice of $b$ is $f^*(m)$.  We have just discovered the convex conjugate $f^*$.  The whole point of $f^*$ is that it tells us how to view $f$ as a supremum of affine functions.  You give $f^*$ a slope $m$, and it gives you the best choice of $b$.
It now becomes very intuitive that $f$ can be recovered from $f^*$, because:
\begin{align}
f(x) &= \sup_m \, \langle m, x \rangle - f^*(m) \quad \text{(because $f$ is a supremum of affine functions)} \\
&= f^{**}(x).
\end{align}
While it is obvious that $f$ can be recovered from $f^*$, the fact that the "inversion formula" $f = f^{**}$ is so simple is a surprising and beautiful fact.
I wrote a similar explanation with some more details here:
Please explain the intuition behind the dual problem in optimization.