Feature selection techniques have been successfully applied in many applications for making supervised learning more effective and efficient. These techniques have been widely used and studied in traditional supervised learning settings, where each instance is expected to have a label. In multiple instance learning (MIL) each example or bag consists of a variable set of instances, and the label is known for the bag as a whole, but not for the individual instances it consists of. Therefore utilizing these labels for feature selection in MIL becomes less straightforward. In this paper we study a new feature subset selection method for MIL called HyDR-MI (hybrid dimensionality reduction method for multiple instance learning). The hybrid consists of the filter component based on an extension of the ReliefF algorithm developed for working with MIL and the wrapper component based on a genetic algorithm that optimizes the search for the best feature subset from a reduced set of features, output by the filter component. We conducted an extensive experimental evaluation of our method on five benchmark datasets and 17 classification algorithms for MIL. The results of our study show the potential of the proposed hybrid with respect to the desirable effect it produces: a significant improvement of the predictive performance of many MIL classification techniques as compared to the effect of filter-based feature selection. This is achieved due to the possibility to decide how many of the top ranked features are useful for each particular algorithm and the possibility to discard redundant attributes.