DPP-HSS: Towards 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 multi-objective 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 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 hypervolume contribution is very time-consuming. To efficiently solve HSS problems while maintaining a good solution quality, we propose a fast and scalable hypervolume subset selection method for many-objective optimization based on the determinantal point process (DPP), named DPP-HSS, which is fully free of hypervolume contribution calculation. Specifically, DPP-HSS constructs a hypervolume 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 hypervolume subset selection algorithm is implemented to solve this transformed problem. Extensive experiments show that the proposed DPP-HSS achieves significant speedup and good hypervolume 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.