Monoidal cut strengthening revisited

There are two distinct strengthening methods for disjunctive cuts with some integer variables; they differ in the way they modularize the coefficients. In this paper, we introduce a new variant of one of these methods, the monoidal cut strengthening procedure, based on the paradox that sometimes weakening a disjunction helps the strengthening procedure and results in sharper cuts. We first derive a general result that applies to cuts from disjunctions with any number of terms. It defines the coefficients of the cut in a way that takes advantage of the option of adding new terms to the disjunction. We then specialize this result to the case of split cuts for mixed integer programs with some binary variables, in particular Gomory mixed integer cuts, and to intersection cuts from multiple rows of a simplex tableau. In both instances we specify the conditions under which the new cuts have smaller coefficients than the cuts obtained by either of the two currently known strengthening procedures.