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 An Arranged Riley Riddle
How to find a minimax solution to a set of inequalities?
Let's consider the following 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 $\max(abs(a), abs(b), abs(c), abs(d), abs(e))$ 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 $eb\ge 9$ 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 $a \in [2.5,4.5]$

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 $eb\ge 9$ 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 $a \in [2.5,4.5]$
20180722 05:21:00