Suppose I have an objective function in my LP as follows

$max$ $|x|$

Based on some googling, I have found there are two ways to convert this into a standard LP.

Method 1.

$|x|$ = $ x^+ + x^-$

$x = x^+ - x^-$

$max$ $x^+ - x^-$

subject to

$x^+, x^- > 0$

Method 2 Since $|x| = max(x,-x)$, we introduce a new variable $t$ and rewrite the problem as

$max$ $t$

subject to

$t >= x$

My main question is are these 2 methods equivalent? Is one of them preferred over the other? $t >= -x$


If your objective function was a minimization problem, then the two methods would work and are equivalent. For instance, consider a problem where $x=-2$. Then, using method 2 for a minimization problem, you would have

$\begin{equation} \text{min }t\\ t \geq 2\\ t \geq -2\\ \end{equation} $

and the solution is 2. Now try it with a max - the solution will be unbounded above.

However, you have a maximization problem. Note that $|x|$ is a convex function. There is no LP reformulation of this problem.


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