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

Neural and Evolutionary Computing;Machine Learning