  We generalize the additive constrained Gaussian process framework to handle
interactions between input variables while enforcing monotonicity constraints
everywhere on the input space. The block-additive structure of the model is
particularly suitable in the presence of interactions, while maintaining
tractable computations. In addition, we develop a sequential algorithm, MaxMod,
for model selection (i.e., the choice of the active input variables and of the
blocks). We speed up our implementations through efficient matrix computations
and thanks to explicit expressions of criteria involved in MaxMod. The
performance and scalability of our methodology are showcased with several
numerical examples in dimensions up to 120, as well as in a 5D real-world
coastal flooding application, where interpretability is enhanced by the
selection of the blocks.
