A (p, q)-Averaged Hausdorff Distance for Arbitrary Measurable Sets

The Hausdorff distance is a widely used tool to measure the distance between different sets. For the approximation of certain objects via stochastic search algorithms this distance is, however, of limited use as it punishes single outliers. As a remedy in the context of evolutionary multi-objective optimization (EMO), the averaged Hausdorff distance Delta(p) has been proposed that is better suited as an indicator for the performance assessment of EMO algorithms since such methods tend to generate outliers. Later on, the two-parameter indicator Delta(p,q) has been proposed for finite sets as an extension to Delta(p) which also averages distances, but which yields some desired metric properties. In this paper, we extend Delta(p,q) to a continuous function between general bounded subsets of finite measure inside a metric measure space. In particular, this extension applies to bounded subsets of R-k endowed with the Euclidean metric, which is the natural context for EMO applications. We show that our extension preserves the nice metric properties of the finite case, and finally provide some useful numerical examples that arise in EMO.