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Among the most famous algorithms for solving classification problems aresupport vector machines (SVMs), which find a separating hyperplane for a set oflabeled data points. In some applications, however, labels are only availablefor a subset of points. Furthermore, this subset can be non-representative,e.g., due to self-selection in a survey. Semi-supervised SVMs tackle thesetting of labeled and unlabeled data and can often improve the reliability ofthe results. Moreover, additional information about the size of the classes canbe available from undisclosed sources. We propose a mixed-integer quadraticoptimization (MIQP) model that covers the setting of labeled and unlabeled datapoints as well as the overall number of points in each class. Since the MIQP'ssolution time rapidly grows as the number of variables increases, we introducean iterative clustering approach to reduce the model's size. Moreover, wepresent an update rule for the required big-$M$ values, prove the correctnessof the iterative clustering method as well as derive tailoreddimension-reduction and warm-starting techniques. Our numerical results showthat our approach leads to a similar accuracy and precision than the MIQPformulation but at much lower computational cost. Thus, we can solve solvelarger problems. With respect to the original SVM formulation, we observe thatour approach has even better accuracy and precision for biased samples.

Mixed-Integer Quadratic Optimization and Iterative Clustering Techniques for Semi-Supervised Support Vector Machines