Find (a, b, c) which minimizes $\max(a, b, c) - min(a, b, c)$ is infeasible?

Question

Is the following objective infeasible ?

$min: f(a, b, c) = \max(a, b, c) - \min(a, b, c)$

In case of fixing the RHS, function could be minimzed by maximizing the LHS.

I have reformlated the problem,

$max: f(a, b, c) = \min(a, b, c)$

$s.t \ \ \ g(a, b, c) = max(a, b, c), \ $ $is \ minimum$

In many cases, I can't take a step through maximizing $f$ without violating $g$.

I guess there's something wrong with my assumptions, if so what it is?

I am not sure if I get what you mean. Isn’t $a=b=c$ minimize that function? — acat3, Aug 15 '20 at 11:26
@Rezha Adrian Tanuharja, No they have to be different, say a set of numbers, they should have a different indices. — , Aug 15 '20 at 11:28

score 0 · Accepted Answer · answered Aug 15 '20 at 11:35

Clearly, the objective function is nonnegative.

Since you impose the constraint that the $3$ inputs are distinct, they must be positive.

Let $c=0, b = \frac{\epsilon}2, a = \epsilon$ where $\epsilon > 0$, we can see that the problem doesn't have a minimum as $\epsilon$ can be made arbitrarily small.

Find (a, b, c) which minimizes $\max(a, b, c) - min(a, b, c)$ is infeasible?

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