Multi-Objective Optimization Problem With Hardly Dominated Boundaries: Benchmark, Analysis, and Indicator-Based Algorithm

The hardly dominated boundary (HDB) is commonly observed in multi-objective optimization problems (HDBMOPs). However, there are only a few benchmark problems related to HDB-MOPs in the evolutionary computation community, which is insufficient to validate the performance of multi-objective evolutionary algorithms (MOEAs). In this paper, we first introduce a new set of HDB-MOPs characterized by various shapes of Pareto fronts and scalable HDB sizes. We then systematically analyze the capabilities of several representative existing MOEAs in handling HDB-MOPs and reveal their strengths and weaknesses in solving this type of problem. Finally, based on this insightful analysis, we propose an indicator-based MOEA with an adaptive reference point to effectively address HDB-MOPs. The source codes of the proposed benchmark problems and the IMOEAARP algorithm are available from https://github.com/CIAMGroup/ EvolutionaryAlgorithm Codes/tree/main/IMOEA-ARP.