antennense

2022-07-15

Let's consider the following linear inequalities:
$a-10\le b\le a-7\phantom{\rule{0ex}{0ex}}b+3\le c\le b+6\phantom{\rule{0ex}{0ex}}c+3\le d\le c+6\phantom{\rule{0ex}{0ex}}d+3\le e\le d+6$
Is there a way to find a solution to this system where $max\left(|a|,|b|,|c|,|d|,|e|\right)$ is minimized?

lywiau63

Expert

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 $max\left(|b|,|e|\right)\ge 4.5$
An optimal solution is $\left(2.5,-4.5,-1.5,1.5,4.5\right)$ and others are similar with $a\in \left[2.5,4.5\right]$

dream13rxs

Expert