If we have multiple absolute values, how do we proceed to solve the problem by reformulating as a linear programming model?
F.ex assume we have the following problem:
Minimize $x_1 + 2|2x_2 + 5|$
Subject to: $|x_1 + 4| - |4x_2| \le 7$
In first glance, i would assume we would need to assign x, t, z for the positive and negative values, and reformulate them under the constraint , but not completely sure - any help would be greatly appreciated!
You cannot solve it as an LP because the feasible region is not convex. For example, $(4,1)$ and $(4,-1)$ are both feasible, but $(4,0)$ is not.
But you can consider the two cases $x_2 \ge 0$ and $x_2 \le 0$ separately, avoiding the $|4x_2|$. Each of these two cases can be linearized, with only one additional variable, and solved as an LP. One yields objective value $-1$, and the other yields objective value $-21$, so the smaller value $\min(-1,-21)=-21$ is optimal.