Original Optimization:
Let say I want to minimize below function:
minimize
$\left|a_1x_1+a_2x_2-c_1\right| + \left|b_1x_1+b_2x_2-c_2\right|$
subject to:
$x_1^{lower} \le x_1 \le x_1^{upper}$
$x_2^{lower} \le x_2 \le x_2^{upper}$
Solutions online: I have seen that the proposed solutions are mainly to introduce an auxiliary variables $t_1$ and $t_2$, and try to minimize $(t_1 + t_2)$ subject to some constraints around $t_1$ and $t_2$ values. (e.g. this post).
Can below Linear Programming reformulation work?
But here is another possibility with seems simpler and effective. I am not sure what am I missing and why nobody has suggested this:
maximize
$a_1x_1+a_2x_2 + b_1x_1+b_2x_2$
subject to:
$a_1x_1+a_2x_2 \le c_1$
$b_1x_1+b_2x_2 \le c_2$
$x_1^{lower} \le x_1 \le x_1^{upper}$
$x_2^{lower} \le x_2 \le x_2^{upper}$
No, it doesn't. Let's consider an extreme case where $$\min |x_1 +x_2 - 1|+ |x_1 +x_2 - 1|$$
$$1 \le x_1 \le 1$$
$$1 \le x_2 \le 1$$
which is feasible.
Your method would construct the following feasible set.
$$x_1 + x_2 \le 1$$ $$x_1 + x_2 \le 1$$ $$1 \le x_1 \le 1$$
$$1 \le x_2 \le 1$$
which is not feasible.