Sparse Multi-term Disjunctive Cuts for the Epigraph of a Function of Binary Variables

We propose a new method for generating cuts valid for the epigraph of a function of binary variables. The proposed cuts are disjunctive cuts defined by many disjunctive terms obtained by enumerating a subset I of the binary variables. We show that by restricting the support of the cut to the same set of variables I, a cut can be obtained by solving a linear program with 2(vertical bar I vertical bar) constraints. While this limits the size of the set I used to define the multi-term disjunction, the procedure enables generation of multi-term disjunctive cuts using far more terms than existing approaches. Experience on three MILP problems with block diagonal structure using vertical bar I vertical bar up to size 10 indicates the sparse cuts can often close nearly as much gap as the multi-term disjunctive cuts without this restriction and in a fraction of the time.