Summary: We present two ways of using cardinality constraints in a decomposition scheme. The first part of the paper describes how one can use cardinality constraints when solving a decomposable linear programming problem. The reason for introducing cardinality constraints in the master problem of the decomposition method is that the successively generated solutions all have the theoretical property that at most as many variables as there are constraints are nonzero in an optimal solution. We will present the effect that the introduction of cardinality constraints will have on the subproblems. In a small numerical example we will also show that the columns generated will differ from the columns generated by the standard Dantzig-Wolfe method.
The second part of the paper discusses a truly cardinality constrained decomposition problem. Here we study a column generation method where practical considerations lead to the necessity to introduce cardinality constraints in the master problem.