Generalized Mixed Integer Rounding Valid Inequalities

We present new families of valid inequalities for (mixed) integer programming (MIP) problems. These valid inequalities are based on a generalization of the 2-step mixed integer rounding (MIR), proposed by Dash and Günlük (2006). We prove that for any positive integer n, n facets of a certain (n + 1)-dimensional mixed integer set can be obtained through a process which includes n consecutive applications of MIR. We then show that for any n, the last of these facets, the n-step MIR facet, can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and Günlük (2006) are simply the first two families corresponding to n = 1, 2, respectively. The n-step MIR inequalities are easily produced using closed-form periodic functions, which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Moreover, for any n, the n-step MIR inequalities define new families of two-slope facets for the finite and the infinite cyclic group problems.