Cutting planes from a mixed integer Farkas lemma

The beauty of linear programming theory arises from linear programming duality that provides a short certificate that a given pair of primal and dual feasible solutions are optimal. One of the central fundaments of this duality result is the so-called Farkas Lemma that establishes that a given system of linear inequalities has a solution if and only if another system of linear inequalities is inconsistent. Unlike in linear programming an integer programming problem is not expected to have a short certificate for optimality. An integer programming duality in the vein of linear programming does not exist. The reason is that there is no integer Farkas Lemma available when we require integrality and simultaneously bounds for the variables. However for an integer programming problem with unbounded variables, i.e., the question of deciding whether a given point belongs to a lattice, a Farkas-type lemma exists, due to Edmonds and Giles [2]. However, to the best of our knowledge, it has not