A Generalization of the Averaged Hausdorff Distance

The averaged Hausdorff distance Delta(p) is an inframetric which has been recently used in evolutionary multiobjective optimization (EMO). In this paper we introduce a new two-parameter performance indicator Delta(p,q) which generalizes Delta(p) as well as the standard Hausdorff distance. For p, q >= 1 the indicator Delta(p,q) (that we call the (p, q)-averaged distance) turns out to be a proper metric and preserves some of the Delta(p) advantages. We proof several properties of Delta(p,q), and provide a comparison with Delta(p) and the standard Hausdorff distance. For simplicity we restrict ourselves to finite sets, which is the most common case, but our results can be extended to the continuous case.