2
$\begingroup$

I am having trouble writing the following optimization problem as a linear program (LP)

$$\min_{x \in \mathbb R^2} \big| | x_{1} - a_{1} | - | x_{2} - a_{2} | \big|$$

where $a \in \mathbb Z^2$ is given. I tried adding the following constraints and replacing objective function to min $U$, but it did not work out.

$$x_{i}-a_{i}\le y_{i}$$

$$a_{i}-x_{i}\le y_{i}$$

$$y_{1}-y_{2}\le U_{}$$

$$y_{2}-y_{1}\le U_{}$$

$\endgroup$
1
$\begingroup$

This sounds like a homework problem, so just hints here.

We'd usually just transform $x_1$ and $x_2$ into two new variables, $$ \begin{array}{ccccc} x_1^{'}~{\equiv}~x_1-a_1 & & \text{and} & & x_2^{'}~{\equiv}~x_2-a_2 \end{array} , $$such that the new problem is$$ \min{\left| \begin{array}{c} {\left|x_1^{'}\right|}-{\left|x_2^{'}\right|} \end{array} \right|}. $$ Then, the absolute values imply symmetries such that we don't need to consider some parts of the domain.

We may ignore:

  1. $x_1^{'}<0$, because it's degenerate with $x_1^{'}>0$ under $\left|x_1^{'}\right|$.

  2. $x_2^{'}<0$, because it's degenerate with $x_2^{'}>0$ under $\left|x_2^{'}\right|$.

  3. $\left|x_1^{'}\right|<\left|x_2^{'}\right|$, because it's degenerate with $\left|x_1^{'}\right|>\left|x_2^{'}\right|$ under $\left|\left|x_1^{'}\right|-\left|x_2^{'}\right|\right|$.

Typically you can specify these sorts of domains to ignore in terms of constraints.

Then, since the degenerate cases have been removed, the objective function's just$$ \min{ \left( {x_1^{'}}-{x_2^{'}} \right) }. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.