Alternative formulations of mixed integer programs

We study the formation of mixed-integer programming (MIP) constraints through the development of constructions which syntactically parallel the set operations of union, intersection, Cartesian product, and linear affine transformation. In this manner, we are able to both modularize the work of constructing representations (as the representations for base sets of a composite construction need not be disjunctively derived) and to make connections to certain of the logic-based approaches to artificial intelligence which utilize the intersection (“and”) and union (“or”) operations. We provide results which allow one to “calculate” the linear relaxation of a composite construction, in terms of set operations on the relaxation of the base sets. We are also able to compare the size of the relaxations for different formulations of the same MIP set, when these different formulations arise from one another through distributive laws. Utilizing these results, we generalize the Davis-Putnam algorithm of propositional logic to an MIP form, and answer a question regarding the relative efficiency of two versions of this algorithm. In this context, the subroutines of the logic algorithm correspond to list processing subroutines for MIP to be used prior to running linear programs. They are similar in nature to preprocessing routines, wherein entire MIP constraint sets are manipulated as formal symbols of logic.