Absolute value in Linear programming problem

Question

I have this problem, where I have to reformulate it as a linear programming problem.

Minimize $ 2x_1 +3 \vert {x_2-10}\vert $

subject to $\vert {x_1+2}\vert + \vert {x_2}\vert ≤5 $

The subject is new to me. Can anyone help?

Answer 1

Alternatively, the constraint $\vert {x_1+2}\vert + \vert {x_2}\vert ≤5$ corresponds to: $$\pm(x_1+2)\pm x_2 \le 5 \iff \begin{cases}\ \ \ \ \ x_1+2+x_2\le 5\\ -(x_1+2)+x_2\le 5\\ \ \ \ \ \ x_1+2-x_2\le 5\\ -(x_1+2)-x_2\le 5\\ \end{cases}$$ The feasible region is:

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Since $x_2<10$, then the original problem can be reformulated as: $$\text{Minimize} \ \ 2x_1+3(10-x_2) \ \ \text{subject to:}\\ \begin{cases}x_1+x_2\le 3\\ x_1-x_2\ge -7\\ x_1-x_2\le 3\\ x_1+x_2\ge -7\\ \end{cases}$$ The answer is: $f(-2,5)=11$. To be verified: Original and Reformulated.