Recent work has shown substantial performance improvements of discriminative probabilistic models over their generative counterparts. However, since discriminative models do not capture the input distribution of the data, their use in missing data scenarios is limited. To utilize the advantages of both paradigms, we present an approach to train Gaussian mixture models (GMMs) in a hybrid generative-discriminative way. This is accomplished by optimizing an objective that trades off between a generative likelihood term and either a discriminative conditional likelihood term or a large margin term using stochastic optimization. Our model substantially improves the performance of classical maximum likelihood optimized GMMs while at the same time allowing for both a consistent treatment of missing features by marginalization, and the use of additional unlabeled data  in a semi-supervised setting. For the covariance matrices, we employ a diagonal plus low-rank matrix structure to model important correlations while keeping the number of parameters small. We show that a non-diagonal matrix structure is crucial to achieve good performance and that the proposed structure can be utilized to considerably reduce classification time in case of missing features. The capabilities of our model are demonstrated in extensive experiments on real-world data.