Minimize Objective Containing a min Operator

Note

This article was originally posted to the AIMMS Tech Blog.

This article explains how to minimize an objective which contains a binary min operator.

The idea of using this binary min operator was to introduce a ceiling to the contribution of a variable to the objective.

For example, let’s take a minimization objective containing the following term:

1
50 * min( p_par, v_var).

Adding parameters and variables

If you want to keep your problem linear, you cannot use the min operator directly in a constraint, so you will have to work around it with some modelling. You can do this introducing a new variable min_param_var, that will be forced to have a value equal to the binary min operator by using additional constraints. This new variable will replace the original min( p_par, v_var) term in the objective (and thus will have coefficient of 50 in objective in the above example).

We also need a new binary variable param_smaller_than_var, with the value 1 if the value of the parameter is smaller than the value of the variable, and 0 otherwise.

Big M and constraints

Finally, we need to add two constraints and a Big M to the model to ensure the newly introduced variables get the correct values. The value of the Big M should be a sufficiently large value, which in this particular case is max( p_par, v_var.upper ).

The two constraints to add are:

  1. \(\mathrm{v\_min\_param\_var} \ge \mathrm{p\_par} * \mathrm{v\_param\_smaller\_than\_var}\)
  2. \(\mathrm{v\_min\_param\_var} \ge \mathrm{v\_var} - \mathrm{v\_param\_smaller\_than\_var} * M\)

If the variable v_min_param_var has a positive component in the objective, the solver will try to minimize the value of the variable v_min_param_var and the above constraints will ensure that the variable v_param_smaller_than_var will automatically get the value 1 if p_par <= v_var, and 0 otherwise.

By filling in some values for the parameter and the variable in the above constraints, you can verify that the binary variable must indeed get the correct value to ensure that the v_min_param_var variable will get the smallest value possible.

Adaptation for a max operator

In case you are minimizing an objective that contains a binary max operator, you can use a similar approach. However, in this case you do not need the Big M, but only one additional variable v_max_param_var and these two constraints:

  1. \(\mathrm{v\_max\_param\_var} \ge \mathrm{p\_par}\)
  2. \(\mathrm{v\_max\_param\_var} \ge \mathrm{v\_var}\)

Further reading

Recently, Erwin Kalvelagen discussed a similar topic in his max tricks blog post.

Last Updated: October, 2019