Reference Point Set Specification for IGD and IGD+ Using Learned Distance Schemes in Riesz S-Energy-Based Iterative Replacement Subset Selection
Academic Article in Scopus
The Inverted Generational Distance (IGD) and its variant IGD+ are widely used quality indicators (QIs) for benchmarking multi-objective evolutionary algorithms (MOEAs). A known drawback of these QIs is their strong dependence on a user-defined parameter: the reference point set. Different reference point sets can lead to substantially different results, making their appropriate specification crucial for fair MOEA performance comparisons. Recently, Riesz s -energy-based subset selection methods have shown promise for generating high-quality reference point sets from dense Pareto front approximations. However, existing studies have been limited to the exclusive use of the Euclidean distance within the Riesz s -kernel throughout the subset selection process. In this work, we propose an offline generation hyper-heuristic framework to automatically learn distance schemes and investigate their impact on Riesz s -energy-based subset selection. Essentially, the learned schemes allow different distance functions to be employed at different stages of the subset selection process to produce high-quality reference sets across varying configurations. Validation experiments over multiple Pareto front geometries, subset sizes, and QIs demonstrate that the learned distance schemes generalize better than single-distance schemes and show that schemes consisting only of Euclidean distance are not always preferable. Furthermore, an application study shows that the resulting reference point sets better approximate the pReferences induced by IGD and IGD+ when compared against state-of-the-art reference point set generation methods. Overall, this work highlights distance scheme selection as a critical component in Riesz s -energy-based subset selection and provides a reusable, learning-based approach for constructing high-quality reference point sets in evolutionary multi-objective optimization.
