Multi-instance learning by maximizing the area under receiver operating characteristic curve

The purpose of this study is to solve the multi-instance classification problem by maximizing the area under the Receiver Operating Characteristic (ROC) curve obtained for witness instances. We derive a mixed integer linear programming model that chooses witnesses and produces the best possible ROC curve using a linear ranking function for multi-instance classification. The formulation is solved using a commercial mathematical optimization solver as well as a fast metaheuristic approach. When the data is not linearly separable, we illustrate how new features can be generated to tackle the problem. We present a comprehensive computational study to compare our methods against the state-of-the-art approaches in the literature. Our study reveals the success of an optimal linear ranking function through cross validation for several benchmark instances.