I have a problem that is almost the typical in linear programming, but not quite. All variables take real non-negative values. Certain simple linear inequalities and equalities should hold. But what does not fit (at least, at first) the typical LP problem is the goal: I want to minimize the number of variables taking strictly positive values.
Is there a trick to handle this as a simple LP problem? Should this be done with mixed integer linear programming? If not, how would you try to solve it?