Multi-objective Evolutionary Algorithm with Evolutionary-status-driven Environmental Selection

The hardly dominated boundary (HDB) is a common feature of multi-objective optimization problems (MOPs). Previous studies have proposed several multi-objective evolutionary algorithms (MOEAs) to deal with the problem characterized by HDBs. Nevertheless, these methods lack consideration of the evolutionary status, potentially resulting in low effeciency when addressing problems that entail a confluence of HDB and other intricate characteristics. This paper proposes an MOEA with evolutionary-status-driven environmental selection (MOEA-ESD) to address such an issue. Specifically, in each generation, the current and historical information are used to evaluate the evolutionary status. Based on the estimated evolutionary status, the proposed MOEA-ESD algorithm adaptively uses three types of environmental selection: the traditional environmental selection (TES) based on Pareto-dominance and crowding distance, a convergence-first environmental selection (CFES) based on Gaussian mixture model clustering, and a diversity-first environmental selection (DFES) based on outlier detection and extreme solutions. In this way, inferior solutions situated on the HDB can be effectively eliminated, thereby facilitating the convergence of the population while concurrently preserving a commendable degree of diversity. Moreover, a set of new benchmark problems with different objective magnitudes and complicated Pareto sets is developed to enrich the features of HDB-MOPs and to verify the algorithm's performance. Our experimental results on 22 HDB-MOPs show the promising performance of the proposed algorithm. The source code of MOEA-ESD is available at https://github.com/CIAM-Group/EvolutionaryAlgorithm_Codes/tree/main/MOEA-ESD.