Why is
\begin{equation}
\begin{aligned}
& \min\max_{i=1,\ldots,n}
& &a_i^Tx+b_i\\
\end{aligned}
\end{equation}
equivalent to an LP
\begin{equation}
\begin{aligned}
& \min
& &t\\
& \text{s.t.} & & a_i^Tx+b_i\leq t \ \ \ \ \ \ \ i = 1,\ldots, n\\
\end{aligned}
\end{equation}
?
I am confused about in the equivalent LP form, where is "max"? How to say both are equivalent if ignoring "max" in the constraint in the LP?
Best Answer
From here
For example,
is equivalent to:
Note that
$$f(x) = 3x-4 \iff x \ge 3$$
So if we have $x = 5$, then $f(5) = 11$ and
$$11 \le t$$
$$9 \le t$$