A New Framework for Visualising Diversity Properties of Pareto Front Approximations in Many-objective Optimisation

Effective visualisation techniques for Pareto front (PF) approximations are crucial in understanding and developing new methodologies for many-objective optimisation (MaOP) analysis. However, the state-of-the-art visualisation methods in the literature fail to reveal the regional diversity of PF properties in a high-dimensional objective space. A new method named Cardinality Distribution (CD) is proposed in this paper, which reveals PF approximations by analysing the cardinality distribution of PF solutions in the objective space. The PF solutions are grouped in a sequence of rings (in three-dimensional space, 3D) or hyper-rings (in higher-dimensional space, mD) in the objective space, where cardinalities are calculated and compared to what they would be if the PF solutions thoroughly and evenly covered the space. CD may detect fully or partially covered or degenerated PF approximations more effectively than state-of-the-art methods. MaF benchmark problems are used to demonstrate the effectiveness of the proposed method.