I was reading about the following trick in convex analysis. Consider the (not obvious) LP
$$p = \max_i \ a_i^Tx + b_i$$
but apparently it can be equivalently reformulated as an LP:
$$ \min_{x,t} \ t, \quad s.t. \ a_i^Tx + b_i \leq t$$
Now, I cannot figure out how you go from one into the other. Isn't $p = \lVert Ax + b \rVert_\infty$?
Also, I am not even sure what the index $i$ in $\max_i$ is supposed to mean - searching an index in a matrix $A$ and vector $b$ that maximizes $a_i^Tx + b_i$?
Given a finite number of reals $r_1,r_2,\ldots,r_n$. For any other real number $t$, the following 2 conditions are equivalent:
(a) The largest of the $r_i$'s is not bigger than $t$.
(b) Neither of the $r_i$'s is bigger than $t$.
Thus, the smallest $t$ satisfying condition (b) is precisely the largest of the $r_i$'s, that is, $$\max\{r_i | i=1,2,\ldots,n\} = \min\{t | r_1 \le t, r_2 \le t, \ldots, r_n\le t\}.$$