I have a model where there is a constraint on the sum of absolute values, and I would like to add a penalty on the shortfall from the constraint. More specifically:
\begin{align*} \text{maximize}\ & \sum a_i x_i \\ \text{subject to}\ & y_i \ge x_i \\ & y_i \ge -x_i \\ &\sum y_i \le \text{ABSLIMIT}, \end{align*} and various linear constraints among subsets of the $x$'s. ($\text{ABSLIMIT}$ is a constant.)
I would like to add a penalty to the objective that would accomplish this:
$$\text{maximize}\ \sum a_ix_i - K\left(\text{ABSLIMIT} - \sum |x_i|\right)$$
In other words, when the $\text{ABSLIMIT}$ constraint is not binding, I would like a penalty which would force the magnitudes of the $x_i$ to be as large as is feasible.
Any suggestions or references would be appreciated. Thanks.