Background: I was reading the following paper: A first-order primal-dual algorithm for convex problems with applications to imaging. At the beginning of the paper, they state the following:
The general problem we consider in this paper is the generic saddle-point problem $\begin{align}\min_{x\in X} \max_{y\in Y} \langle Kx, y\rangle + G(x)-F^*(y) \end{align}\tag*{} $ where $G: X\to [0,+\infty)$ and $F^*: Y\to [0, +\infty)$ are proper, convex, lower-semicontinuous (l.s.c.) functions, $F^*$ being itself the convex conjugate of a convex l.s.c. function $F$.
The word "the saddle-point problem" was unfamiliar to me, so I looked up Google and found the following: Introduction to Saddle Point Problems.
Summarizing the pdf, I understood the following:
For a convex optimization problem:
$\begin{align} &\textrm{minimize } J(u) = \frac{1}{2}\langle Au, u\rangle - \langle f, u \rangle \\ &\textrm{subject to } Bu = g \end{align}$
where $A$ is a symmetric $n\times n$ matrix, $B$ is a $m\times n$ matrix, $f\in \mathbb{R}^n,g\in \mathbb{R}^m$,
The Lagrangian is defined by $\mathcal{L}(u, p) =\frac{1}{2}\langle Au, u\rangle - \langle f, u\rangle +\langle p, Bu-g \rangle$
By the KKT condition, the saddle points $(u^*, p^*)$ of the Lagrangian are the solutions to $\min_{u}\max_{p} \mathcal{L}(u,p)$.
The last min-max problem is somewhat close to the min-max problem that I saw in the paper, so I am fairly certain they are related, but I cannot quite make a connection between the two min-max problems. So my question is the following.
Why is the first min-max problem called the saddle-point problem? How is it related to the second min-max problem?