Binary extended formulations and sequential convexification

A binarization of a bounded variable x is a linear formulation with variables x and additional binary variables y1, . . . , yk, so that integrality of x is implied by the integrality of y1, . . . , yk. A binary extended formulation of a polyhedron P is obtained by adding to the original description of P binarizations of some of its variables. In the context of mixed-integer programming, imposing integrality on 0/1 variables rather than on general integer variables has interesting convergence properties and has been studied both from the theoretical and from the practical point of view. We propose a notion of natural binarizations and binary extended formulations, encompassing all the ones studied in the literature. We give a simple characterization of the vertices of such formulations, which allows us to study their behavior with respect to sequential convexification. In particular, given a binary extended formulation and one of its variables x, we study a parameter that measures the progress made towards ensuring the integrality of x via application of sequential convexification. We formulate this parameter, which we call rank, as the solution of a set covering problem and express it exactly for the classical binarizations from the literature.