  We introduce an additive Gaussian process framework accounting for
monotonicity constraints and scalable to high dimensions. Our contributions are
threefold. First, we show that our framework enables to satisfy the constraints
everywhere in the input space. We also show that more general componentwise
linear inequality constraints can be handled similarly, such as componentwise
convexity. Second, we propose the additive MaxMod algorithm for sequential
dimension reduction. By sequentially maximizing a squared-norm criterion,
MaxMod identifies the active input dimensions and refines the most important
ones. This criterion can be computed explicitly at a linear cost. Finally, we
provide open-source codes for our full framework. We demonstrate the
performance and scalability of the methodology in several synthetic examples
with hundreds of dimensions under monotonicity constraints as well as on a
real-world flood application.
