Improving Robustness of Stopping Multi-objective Evolutionary Algorithms by Simultaneously Monitoring Objective and Decision Space

Appropriate stopping criteria for multi-objective evolutionary algorithms (MOEA) are an important research topic due to the computational cost of function evaluations, particularly on real-world problems. Most common stopping criteria are based on a fixed budget of function evaluations or the monitoring of the objective space. In this work, we propose a stopping criterion based on monitoring both the objective and decision space of a problem. Average Hausdorff distance (AHD) and genetic diversity are used, respectively. Two-sided t-tests on the slope coefficients after regression analyses are used to detect the stagnation of the AHD and the genetic diversity. The approach is implemented for two widely used MOEAs: NSGA-II and SPEA2. It is compared to a fixed budget, the online convergence detection approach, and the individual monitoring of each space on four bi-objective and two three-objective benchmark problems. Our experimental results reveal that the combined approach achieved significantly better results than the approaches considering only one of the spaces. In particular, we find that the combined consideration runs longer and hence more robustly ensures a well-approximated Pareto front. Nevertheless, on average 29% and 17% function evaluations are saved for NSGA-II and SPEA2, respectively, compared to standard budget recommendations.