I am trying to reformulate a linear programming problem with an absolute value in the objective function. I decided to use a proxy method mentioned in the 'Absolute values in the objective function' section of this article.
I didn't get why 'if the objective is a minimization problem of the form $-|f(x)| + g(y)$ or is a maximization problem of the form $|f(x)| + g(y)$, unfortunately, the model cannot be reformulated as a standard LP model'.
For example, if we look at a maximization problem of the form $|f(x)| + g(y)$, can we not reformulate this as:
$\max U + g(y)$
s.t. $U \le |f(x)|$, i.e. $U\le f(x)$ and $U\le-f(x)$
I don't see why this formulation wouldn't work?
Best Answer
It does not work for maximizing $|f(x)|+g(y)$ because $U \leq f(x)$ and $U \leq -f(x)$ together with the objective of maximizing $U$ means that at optimality $U = \min\{f(x), -f(x)\} = -|f(x)|$. So you are maximizing $-|f(x)|+g(y)$ instead of $|f(x)|+g(y)$.