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# How to find a minimax solution to a set of inequalities?

2018-07-22 02:47:48

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 $e-b\ge 9$ 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 $a \in [2.5,4.5]$

• 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 $a \in [2.5,4.5]$

2018-07-22 05:21:00