How to find a minimax solution to a set of linear inequalities? Let's consider the following linear inequalities:
$$a - 10 \leq b \leq  a - 7 \\
b + 3  \leq c \leq  b + 6 \\
c + 3  \leq d \leq  c + 6 \\
d + 3  \leq e \leq  d + 6$$
Is there a way to find a solution to this system where $\max \big( |a|, |b|, |c|, |d|, |e| \big)$ is minimized?

With the changed question, the minimised maximum absolute value is $4.5$
You have $e \ge d+3 \ge c+6 \ge b+9$ so $e-b\ge 9$ and $\max(|b|,|e|) \ge 4.5$
An optimal solution is $(2.5,-4.5,-1.5,1.5,4.5)$ and others are similar with $a \in [2.5,4.5]$ 
