  Artificial Neural Networks (ANNs) have become a powerful tool for modeling
complex relationships in large-scale datasets. However, their black-box nature
poses ethical challenges. In certain situations, ensuring ethical predictions
might require following specific partial monotonic constraints. However,
certifying if an already-trained ANN is partially monotonic is challenging.
Therefore, ANNs are often disregarded in some critical applications, such as
credit scoring, where partial monotonicity is required. To address this
challenge, this paper presents a novel algorithm (LipVor) that certifies if a
black-box model, such as an ANN, is positive based on a finite number of
evaluations. Therefore, as partial monotonicity can be stated as a positivity
condition of the partial derivatives, the LipVor Algorithm can certify whether
an already trained ANN is partially monotonic. To do so, for every positively
evaluated point, the Lipschitzianity of the black-box model is used to
construct a specific neighborhood where the function remains positive. Next,
based on the Voronoi diagram of the evaluated points, a sufficient condition is
stated to certify if the function is positive in the domain. Compared to prior
methods, our approach is able to mathematically certify if an ANN is partially
monotonic without needing constrained ANN's architectures or piece-wise linear
activation functions. Therefore, LipVor could open up the possibility of using
unconstrained ANN in some critical fields. Moreover, some other properties of
an ANN, such as convexity, can be posed as positivity conditions, and
therefore, LipVor could also be applied.
