A question about the extremum of absolute value

Question

For all $a,b,c\in \mathbb{R}$ , how to find the minimum of the function $$f(a,b,c)=\max\{|a+1|,|2a+b|,|b+2c|,|c-2|\}\,? $$

I know that all of them are above $0$, but I don't know how to solve this, since the four values are connected with each other.

Addition: My teacher gave me a answer, which is from @Siong Thye Goh 's answer. Call the biggest one $A$, and trivially $6A \geq 2|a+1|+|2a+b|+|b+2c|+2|c-2| \geq |-2a-2+2a+b-b-2c+2c-4| = 6$ So $A \geq 1$.

Answer 1

I believe there is a simpler approach. Hopefully knowing the answer can help you find a better approach.

This is a problem that can be approached by a tool called linear programming.

You want to solve

$$\min_{a,b,c} \max(|a+1|, |2a+b|, |b+2c|, |c-2|).$$

This can be rewritten as

$$\min z$$

subject to

\begin{align}z &\ge a+1\\ z &\ge -a-1 \\ z &\ge 2a+b \\ z &\ge -2a-b\\ z &\ge b+2c\\ z &\ge -b-2c\\ z &\ge c-2\\ z &\ge -c+2 \end{align}

from scipy.optimize import linprog

c = [1, 0, 0, 0]
A_ub = [[-1, 1, 0, 0], [-1, -1, 0, 0], [-1, 2, 1, 0], [-1, -2, -1, 0], [-1, 0, 1, 2], [-1, 0, -1, -2], [-1, 0, 0, 1], [-1, 0, 0, -1]]
b_ub = [-1, 1, 0, 0, 0, 0, 2, -2]
res = linprog(c, A_ub = A_ub, b_ub = b_ub, method = 'simplex')
print(res)

The solver found that $z=\frac43, a=\frac13, b = 0, c= \frac23$.