Explanation of the maximization in LP formulated "minimax" I am following Prof. Williams's description of the linear
programming formulation [1] of the minimax problem:
Minimize: $\left( \underset{i}{\rm Maximum}
                  \sum\limits_j a_{ij} x_j \right)$
subject to: conventional linear constraints.
Here, {$x_j$} are the decision variables and {$a_{ij}$} are
constant coefficients.  The above is recast into:
Minimize $z$
subject to  $\displaystyle\sum_j a_{ij} x_j - z \le 0$ for all $i$
I get that the search will seek a $z$ that gets as close as possible
to $\displaystyle\sum_i a_{ij} x_j$ because that is the explicit
minimization expression.  What causes $\sum\limits_i a_{ij} x_j$ to be maximized?
[1] Model Building in Mathematical Programming, H. Paul Williams (2013), ed.5, Section 3.2.3 (Minimax objectives), page 27: https://www.researchgate.net/profile/Fazel_Varasteh/post/Can_anybody_please_suggest_a_reference_for_modelling_cost_of_production/attachment/59d63615c49f478072ea3d2f/AS:273636437495809@1442251418466/download/Wiley Model Building in Mathematical Programming 5th       (2013).[sharethefiles.com].pdf
 A: 
I get that the search will seek a $z$ that gets as close as possible to $\sum_i a_{ij} x_j$ because that is the explicit minimization expression.

No, this will find a $z$ that is as small as possible. The objective is "Minimize $z$".

What causes $\displaystyle\sum_i a_{ij} x_j$ to be maximized?

Nothing. There are constraints that must be met, but those are distinct from the expression to be optimized.
A: The following mental picture helped me understand
the effects in play.  It may depend on how people think of things, but
hopefully, it will save someone an afternoon of pondering.
Each $i$ in
$\left( \underset{i}{\rm Maximum} \sum\limits_j a_{ij} x_j \right)$
enumerates an inner product between a row $i$ of array [$a_{ij}$] with the
column vector [$x_j$] of decision variables
Then, $\left( \underset{i}{\rm Maximum} \sum\limits_j a_{ij} x_j \right)$
simply refers to the largest-valued inner product.
We want values
of $x_j$ that yield the smallest possible
$\left( \underset{i}{\rm Maximum} \sum\limits_j a_{ij} x_j \right)$.
In the reformulation using $z$, for each inner product enumerated $i$,
$z$ is forced downward onto the inner product $\sum\limits_j a_{ij} x_j$
from above.  While there are $|\{i\}|$ such constraints, the only one
that matters is the one for which the inner product is largest.  Since
$z$ is made to be as small as possible, the values of $x_j$ will be
found such that the largest inner product is minimized.
