Linear optimization with max function in constraint

Question

I have a linear optimization:

$$ \min_{x_i} -\sum_i x_i $$ $$ \mathrm{s.t.} Ax \le b$$

Where $x$ is a variable vector. I would like to introduce into the linear problem an additional constraint that include the $max$ functions, like this:

$$ \max_i (x_i) + \max_i (-x_i)\le c$$

where $c$ is a given constant. Is there a way to incorporate that constrain keep the linear formulation of the problem?

Thanks in advance for any help or references!

I want something like this, but in my case the max function is in the constraint and not in the optimization function.

Answer 1

Your constraint is equivalent to: $$\max_i x_i - \min_i x_i \leq c$$

You can rewrite it with two more variables $\max X$ and $\min X$ and a linear number of constraints:

$$\max X \geq x_i\;\;\forall i$$ $$ \min X \leq x_i\;\;\forall i$$ $$\max X-\min X \leq c$$