@article{GODARD2026109749,
title = {Constrained adaptive parallelepipedic approximation of the image of a set by a nonlinear function},
journal = {International Journal of Approximate Reasoning},
volume = {197},
pages = {109749},
year = {2026},
issn = {0888-613X},
doi = {https://doi.org/10.1016/j.ijar.2026.109749},
url = {https://www.sciencedirect.com/science/article/pii/S0888613X26001246},
author = {Maël Godard and Luc Jaulin and Damien Massé},
keywords = {Parallelepipeds, Boundary approach, Adaptive method, Constraint validation},
abstract = {This paper proposes a method to compute an adaptive outer approximation of the image of a set by a function. This approximation relies on a cover of the boundary of the image set with parallelepipeds. Since the boundary is generally not a parallelepiped, covering it comes down to enclosing it in an union of parallelepipeds. The fewer parallelepipeds used, the more pessimistic the approximation becomes. If we want to validate that the image set respects some constraints, excessive pessimism in the cover can lead to incorrect conclusions. Increasing the number of parallelepipeds significantly reduces the pessimism, but also increases the computationnal complexity. The method presented here constructs an adaptive covering of the boundary of the image set. The objective is to verify that the image set satisfies given constraints while saving computationnal effort where possible. In the cases where the constraints can not be validated, we are also able to provide a precise detection of the parts of the initial and image sets boundaries responsible for it.}
}