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. In this setting, a classic result of Bell, Doignon, and Scarf states the existence of a lattice-free gradient polyhedron with at most four facets. 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.

无格梯度多面体可用于证明混合整数凸极小化模型的最优性。我们考虑如何在所有级别集都是有界的假设下为具有两个整数变量的无约束模型构造这些多面体。在这种情况下，Bell，Doignon和Scarf的经典结果表明存在不带格子的梯度多面体，该多面体最多具有四个面。我们提出了一种用于创建一系列梯度多面体的算法，每个梯度多面体最多具有四个面，这些面有限地会聚为无晶格的梯度多面体。每次更新都需要不断进行许多梯度评估。我们的更新模仿了梯度下降算法，因此，它针对具有两个整数变量的问题产生了一种梯度下降类型的算法。