DPP-HSS: Toward Fast and Scalable Hypervolume Subset Selection for Many-Objective Optimization

Hypervolume subset selection (HSS) has received significant attention since it has a strong connection with evolutionary multiobjective optimization (EMO), such as environment selection and post-processing to identify representative solutions for decision-makers. The goal of HSS is to find the optimal subset that maximizes the hypervolume (HV) indicator subject to a given cardinality constraint. However, existing HSS algorithms or related methods are not efficient in achieving good performance in high-dimensional objective spaces. This is primarily because HSS problems become NP-hard when the number of objectives exceeds two, and the calculation of HV contribution (HVC) is very time-consuming. To efficiently solve HSS problems while maintaining a good solution quality, we propose a fast and scalable HSS method for many-objective optimization based on the determinantal point process (DPP), named DPP-HSS, which is fully free of HVC calculation. Specifically, DPP-HSS constructs an HV kernel matrix by extracting the convergence and diversity representations of each solution for a given HSS problem. This matrix is then used to build a DPP model. Subsequently, the original HSS problem is reformulated as a new maximization optimization problem based on the constructed model. A greedy DPP-based HSS algorithm is implemented to solve this transformed problem. Extensive experiments show that the proposed DPP-HSS achieves significant speedup and good HV performance in comparison with state-of-the-art HSS algorithms on benchmark problems. Furthermore, DPP-HSS demonstrates very good scalability with respect to the number of objectives.