  Random forests are among the most famous algorithms for solving
classification problems, in particular for large-scale data sets. Considering a
set of labeled points and several decision trees, the method takes the majority
vote to classify a new given point. In some scenarios, however, labels are only
accessible for a proper subset of the given points. Moreover, this subset can
be non-representative, e.g., due to collection bias. Semi-supervised learning
considers the setting of labeled and unlabeled data and often improves the
reliability of the results. In addition, it can be possible to obtain
additional information about class sizes from undisclosed sources. We propose a
mixed-integer linear optimization model for computing a semi-supervised random
forest that covers the setting of labeled and unlabeled data points as well as
the overall number of points in each class for a binary classification. Since
the solution time rapidly grows as the number of variables increases, we
present some problem-tailored preprocessing techniques and an intuitive
branching rule. Our numerical results show that our approach leads to a better
accuracy and a better Matthews correlation coefficient for biased samples
compared to random forests by majority vote, even if only few labeled points
are available.
