  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.
