MILP Penalty Function Only for Negative Values

Question

This is (hopefully) an easy answer but I haven't dealt with this before.

I have a MILP which includes an unbounded, continuous decision variable. However, I generally don't want this decision variable to take negative values, so I want to include a penalty in the objective function where there is zero penalty incurred for a positive value, but for negative values, the penalty is proportional to the absolute value. Basically, I want a value of -100 to be penalized twice as much as -50, and 10x as much as -10.

Right now I have formulated an indicator variable that is 1 if the decision variable is negative, which incurs a penalty; but this penalty is the same for any negative value.

Any suggestions on how to implement this?

Answer 1

If $x$ denotes your free variable, you can penalize the term $f(x)=\max\{-x,0\}$ in your objective function, which you can linearize by replacing it with a variable $y$, and constraints $y\ge -x, y\ge 0$.

Answer 2

You do not need to introduce an indicator variable. Suppose $x$ is your free variable. Introduce nonnegative variables $x^+$ and $x^-$, replace $x$ with $x^+-x^-$ throughout, and penalize $x^-$ linearly in the objective.