A Newton Method for Hausdorff Approximations of the Pareto Front Within Multiobjective Evolutionary Algorithms

A common goal in evolutionary multiobjective optimization is to find suitable finite-size approximations of the Pareto front of a given multiobjective optimization problem. While many multiobjective evolutionary algorithms (MOEAs) have proven to be very efficient in finding good Pareto front approximations, they may need quite a few resources or may even fail to obtain optimal or nearly optimal approximations. Hereby, optimality is implicitly defined by the chosen performance indicator. In this work, we propose a set-based Newton method for the Hausdorff approximations of the Pareto front to be used within MOEAs. To this end, we first generalize the previously proposed Newton step for the performance indicator to treat constrained problems for general reference sets. To approximate the target Pareto front, we propose a particular strategy for generating the reference set that utilizes the data gathered by the evolutionary algorithm during its run. Finally, we show the benefit of the Newton method as a postprocessing step on several benchmark test functions and different base evolutionary algorithms.