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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 Δ 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 Δ p , q has been proposed for finite sets as an extension to Δ p which also averages distances, but which yields some desired metric properties. In this paper, we extend Δ 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.

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Multi-objective evolutionary algorithms are widely used by researchers and practitioners to solve multi-objective optimization problems (MOPs), since they require minimal assumptions and are capable of computing a finite size approximation of the entire solution set in one run of the algorithm. So far, however, the adequate treatment of equality constraints has played a minor role. Equality constraints are particular since they typically reduce the dimension of the search space, which causes problems for stochastic search algorithms such as evolutionary strategies. In this paper, we show that multi-objective evolutionary algorithms hybridized with continuation-like techniques lead to fast and reliable numerical solvers. For this, we first propose three new problems with different characteristics that are indeed hard to solve by evolutionary algorithms. Next, we develop a variant of NSGA-II with a continuation method. We present numerical results on several equality-constrained MOPs to show that the resulting method is highly competitive to state-of-the-art evolutionary algorithms.

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