We consider the problem of solving a family of parametric mixed-integer linear optimization problems where some entries in the input data change. We introduce the concept of cutting-plane layer (CPL), i.e., a differentiable cutting-plane generator mapping the problem data and previous iterates to cutting planes. We propose a CPL implementation to generate split cuts, and by combining several CPLs, we devise a differentiable cutting-plane algorithm that exploits the repeated nature of parametric instances. In an offline phase, we train our algorithm by updating the internal parameters controlling the CPLs, thus altering cut generation. Once trained, our algorithm computes, with predictable execution times and a fixed number of cuts, solutions with low integrality gaps. Preliminary computational tests show that our algorithm generalizes on unseen instances and captures underlying parametric structures.

翻译：我们考虑求解一族参数化混合整数线性优化问题，其中输入数据中的某些条目会发生变化。我们引入割平面层（CPL）的概念，即一个可微的割平面生成器，它将问题数据和先前迭代结果映射为割平面。我们提出一种CPL实现以生成分割割，并通过组合多个CPL，设计了一种可微的割平面算法，该算法利用参数化实例的重复特征。在离线阶段，我们通过更新控制CPL的内部参数来训练算法，从而改变割的生成。训练完成后，我们的算法能够以可预测的执行时间和固定数量的割，计算出低积分间隙的解。初步计算测试表明，我们的算法在未见过的实例上具有泛化能力，并能捕捉潜在的参数化结构。