I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:

max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$

$s.t. $

$|\alpha_j| \le 1$

Where p(x) = 2x if x < 0, x otherwise, and $ \vec c$ is a constant

The p(x) function is what's troubling me, since one can only determine x's sign after an assignment of $\vec \alpha$. How can I remove the function p from the objective and express this objective equivalently as a linear combination of the variables?

Thank you!