Cutting planes and infeasibility certificates from lattice-point-free polyhedra

A central result in the theory of integer optimization states that a system of linear diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, yT A integral such that yT b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, C x ≤ d}. We show that a certificate of integral infeasibility is a linear system with rank(C) variables containing no integral point. The result also extends to the mixed integer setting.