A Method for a Posteriori Identification of Knee Points Based on Solution Density

Many evolutionary algorithms have been proposed and demonstrated to have excellent performance in striking a balance between convergence and diversity in dealing with multiobjective optimization problems. However, little attention has been paid to the decision making stage where a small number of solutions are selected to be presented to the user. It is believed that knee points are considered to be the naturally preferred solutions when no specific preferences are available, because knee solutions incur a large loss in at least one objective to gain a small amount in other objectives. One common issue in the identification of knee points is that some knee points are easily ignored and knees in concave regions are hard to be identified. To resolve these issues, this paper proposes a novel method for knee identification, which first maps the non-dominated solutions to a constructed hyperplane and then divides them into groups, each representing a candidate knee region, based on the density of the solutions projected on the hyperplane. Finally, the convexity and curvature of the candidate knee groups are determined and only those having a strong curvature are kept. The proposed method is empirically demonstrated to be effective in identifying knee points located in both convex and concave regions on three existing test problems and one newly proposed test problem.