  Decision trees are one of the most famous methods for solving classification
problems, mainly because of their good interpretability properties. Moreover,
due to advances in recent years in mixed-integer optimization, several models
have been proposed to formulate the problem of computing optimal classification
trees. The goal is, given a set of labeled points, to split the feature space
with hyperplanes and assign a class to each partition. In certain scenarios,
however, labels are exclusively accessible for a subset of the given points.
Additionally, this subset may be non-representative, such as in the case of
self-selection in a survey. Semi-supervised decision trees tackle the setting
of labeled and unlabeled data and often contribute to enhancing the reliability
of the results. Furthermore, undisclosed sources may provide extra information
about the size of the classes. We propose a mixed-integer linear optimization
model for computing semi-supervised optimal classification trees that cover the
setting of labeled and unlabeled data points as well as the overall number of
points in each class for a binary classification. Our numerical results show
that our approach leads to a better accuracy and a better Matthews correlation
coefficient for biased samples compared to other optimal classification trees,
even if only few labeled points are available.
