I have a piecewise constraint that I am having a hard time converting using big-M modelling. The context is a gym owner that is updating membership costs subject to churn restrictions. The owner can increase or decrease the prices which will have an effect on the churn probability of a customer, $n$. However, if the change in price is small enough, there is no churn. The function looks like:
$ch_n$ =
\begin{cases} 0, & |p^*_n - p_n|\leq \epsilon \\ m_n(p^*_n - p_n), & p^*_n - p_n > \epsilon \end{cases}
Where $ch_n$ is the churn probability of customer n. This is zero is the price increase is less than some threshold epsilon ($p^*_n$ is the new price and the decision variable. $p_n$ is the old price). If the change is greater than epsilon, the churn probability is a linear function of the price change (so $m_n$ is known for each customer).
The difficulty I am having the the presence of the absolute values in the constraint definition.