二维无格梯度多面体的构造
Constructing lattice-free gradient polyhedra in dimension two
摘要：
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水平集有界的假设下，为两个整数变量的无约束模型构造这些多面体。Bell、Doignon和Scarf的一个经典结果指出，在这种情况下存在一个最多有四个面的无格梯度多面体。我们提出了一个算法，用于创建一个序列的梯度多面体，其中每一个最多有四个方面，该梯度收敛到一个无格梯度多面体。每次更新都需要不断进行多次梯度评估。我们的更新模仿梯度下降算法，因此，它产生一个梯度下降型算法的问题，两个整数变量。
英文摘要：
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.
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