  Lattice-free gradient polyhedra can be used to certify optimality for
mixed-integer convex minimization models. We consider how to construct these
polyhedra for unconstrained models with two integer variables under the
assumption that all level sets are bounded. A classic result of Bell, Doignon,
and Scarf states that a lattice-free gradient polyhedron with at most four
facets exists in this setting. We present an algorithm for creating a sequence
of gradient polyhedra, each of which has at most four facets, that finitely
converges to a lattice-free gradient polyhedron. Each update requires
constantly many gradient evaluations. Our updates imitate the gradient descent
algorithm, and consequently, it yields a gradient descent type of algorithm for
problems with two integer variables.
