A generalization of the averaged hausdorff distance

The averaged Hausdorff distance ?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 ?p,q which generalizes ?p as well as the standard Hausdorff distance. For p, q 1 the indicator ?p,q (that we call the (p, q)-averaged distance) turns out to be a proper metric and preserves some of the ?p advantages. We proof several properties of ?p,q, and provide a comparison with ?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.