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Finding a good classifier is a multiobjective optimization problem withdifferent error rates and the costs to be minimized. The receiver operatingcharacteristic is widely used in the machine learning community to analyze theperformance of parametric classifiers or sets of Pareto optimal classifiers. Inorder to directly compare two sets of classifiers the area (or volume) underthe convex hull can be used as a scalar indicator for the performance of a setof classifiers in receiver operating characteristic space.  Recently, the convex hull based multiobjective genetic programming algorithmwas proposed and successfully applied to maximize the convex hull area forbinary classification problems. The contribution of this paper is to extendthis algorithm for dealing with higher dimensional problem formulations. Inparticular, we discuss problems where parsimony (or classifier complexity) isstated as a third objective and multi-class classification with three differenttrue classification rates to be maximized.  The design of the algorithm proposed in this paper is inspired byindicator-based evolutionary algorithms, where first a performance indicatorfor a solution set is established and then a selection operator is designedthat complies with the performance indicator. In this case, the performanceindicator will be the volume under the convex hull. The algorithm is tested andanalyzed in a proof of concept study on different benchmarks that are designedfor measuring its capability to capture relevant parts of a convex hull.  Further benchmark and application studies on email classification and featureselection round up the analysis and assess robustness and usefulness of the newalgorithm in real world settings.

Multiobjective Optimization of Classifiers by Means of 3-D Convex Hull Based Evolutionary Algorithm