Characterizing Pareto Front Approximations in Many-objective Optimization

Abstract

A Pareto Optimal Front (POF) provides the set of optimal trade-off solutions formulti-objective optimization problems. The characteristics of the POF, e.g. continuity, convexity, spread and uniformity are important in the context of decision making and performance assessment. Most of the existing metrics (hypervolume, inverted generational distance, coverage etc.) were originally designed for two or three objective optimization problems with an aim of assessing the quality of non-dominated solutions delivered by various algorithms. The metrics provide little information about the nature of the front and some of them (e.g. hypervolume) are computationally expensive for problems involving large number of objectives. For problems with more than three objectives, existing tools for visualization such as parallel plots, spider/radar plots and scatter plots also offer limited useful information in terms of the nature of the front. In this paper, we introduce an alternative scalar measure of diversity that is suitable for characterizing POF approximations of optimization problems with high number of objectives. The diversity is measured against a Reference Pareto Front, a set of points uniformly spread on the hyperplane with unit intercepts. We also illustrate that the computation of such a metric is a natural extension of decomposition based evolutionary algorithms which attempt to align solutions with the reference directions constructed using the Ideal point and uniformly distributed points on the hyperplane. In particular, the perpendicular distances between the uniformly distributed reference directions and their closest solutions in the nondominated front provides information about the diversity, while the shortest distance of every solution from the hyperplane provides information about the convexity of the POF. The proposed metrics are illustrated using a number of test problems involving up to fifteen objectives.