In this paper, we examine a mixed integer linear programming (MILP) reformulation for mixed integer bilinear problems where each bilinear term involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to linearize the resul...

Graphical models with latent count variables arise in a number of areas. However, standard inference algorithms do not apply to these models due to the infinite support of the latent variables. Winner & Sheldon (2016) recently developed a new technique using probability generating functions (PGFs) to perform efficient, exact inference for certain Poisson latent variable models. However, the met...

We present a version of GMI (Gomory mixed-integer) cuts in a way so that they are derived with respect to a “dual form” mixed-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. This follows the general scheme of He and Lee, who did the case of Gomory pure-integer cuts. Our input mixed-integer problem is not in standard form,...

Branching in mixed-integer (or integer) linear programming requires choosing both the branching variable and the branching direction. This paper develops a number of new methods for making those two decisions either independently or together with the goal of reaching the first integer-feasible solution quickly. These new methods are based on estimating the probability of satisfying a constraint...

A standard trick in integer programming is to replace each bounded integer-constrained variable with a small number of binary variables, using the bit representation of the given variable. We show that, in the case of mixed-integer quadratic programs (MIQPs), this process can enable one to obtain stronger linear programming relaxations. Moreover, we give a simple sufficient condition under whic...

We show that a 2-variable integer program, defined by m constraints involving coefficients with at most φ bits can be solved with O(m+ φ) arithmetic operations on rational numbers of size O(φ). This result closes the gap between the running time of two-variable integer programming with the sum of the running times of the Euclidean algorithm on φ-bit integers and the problem of checking feasibil...