# Compensating Data Shortages in Manufacturing with Monotonicity Knowledge
## 1 Introduction
Among other things, it is shown in the mentioned articles that, in spite of noise in the output data, the arising monotonized models are close to the true relationship for sufficiently large training datasets. Summarizing, these articles show that for large datasets, noise in the output data can be compensated by monotonization to a certain extent. In contrast to that, in some works, such as [13; 17; 26; 27; 28; 29], monotonicity knowledge was incorporated already in training. In these articles, the monotonicity requirements were added as constraints--either hard [17; 26; 28; 29] or soft [13; 26]--to the data-based optimization of the model parameters. In [13; 28], probabilistic monotonicity notions are used. In [26; 27; 28; 29], support vector regressors in the linear-programming or the more standard quadratic-programming form, Gaussian process regressors or neural network models were considered, and the monotonicity of these models was enforced by constraints on the model derivatives at predefined sampling points [26; 28; 29] or on the model increments between predefined pairs of sampling points [27]. A disadvantage of the projection- and rearrangement-based approaches [22; 23; 24] from the point of view of manufacturing applications is that these methods are tailored to large datasets. Another disadvantage of these approaches is that the resulting models typically exhibit distinctive kinks, which are almost always unphysical. Additionally, the models resulting from the multidimensional rearrangement method by [23] are not guaranteed to be monotonic when trained on small datasets. A drawback of the tilting approach from [21] is that it is formulated and validated only for one-dimensional input spaces (intervals in \(\mathbb{R}\)). Accordingly, naively extending the non-adaptive discretization scheme from [21] to higher dimensions would result in long computation times. A downside of the in-training methods from [26; 28; 29] is that the sampling points at which the monotonicity constraints are imposed have to be chosen in advance (even though they need not coincide with the training data points). With the method proposed in the present article, we address the aforementioned issues and shortcomings. In Section 2, our methodology for monotonic regression using semi-infinite optimization is introduced. It incorporates the monotonicity knowledge during training. Specifically, polynomial regression models are assumed for the input-output relationships to be learned. Since there is no after-training monotonization step in the method, our models are smooth, and in particular, do not exhibit kinks. Also, due to the employed adaptive discretization scheme, the method is computationally efficient also in higher dimensions. To our knowledge, such an adaptive scheme has not been applied to solve monotonic regression problems before, especially not in situations with sparse data. In Section 4, the method is validated by means of two applications to real-world processes which are both introduced in Section 3, namely, laser glass bending and press hardening of sheet metal. It turns out that the adaptive semi-infinite optimization approach to monotonic regression is better suited for the considered applications with their small datasets and the resulting models are more accurate than those obtained with the comparative approaches from the literature.