Guo Yu, Yaochu Jin, Markus Olhofer, Qiqi Liu, Wenli Du, "Solution Set Augmentation for Knee Identification in Multiobjective Decision Analysis", IEEE Transactions on Cybernetics, vol. 53, no. 4, pp. 2480-2493, 2023.

In multiobjective decision making, most knee identification algorithms implicitly assume that the given solutions are well distributed and can provide sufficient information for identifying knee solutions. However, this assumption may fail to hold when the number of objectives is large or when the shape of the Pareto front is complex. To address the above issues, we propose a knee-oriented solution augmentation (KSA) framework that converts the Pareto front into a multimodal auxiliary function whose basins correspond to the knee regions of the Pareto front. The auxiliary function is then approximated using a surrogate and its basins are identified by a peak detection method. Additional solutions are then generated in the detected basins in the objective space and mapped to the decision space with the help of an inverse model. These solutions are evaluated by the original objective functions and added to the given solution set. To assess the quality of the augmented solution set, a measurement is proposed for the verification of knee solutions when the true Pareto front is unknown. The effectiveness of KSA is verified on widely used benchmark problems and successfully applied to a hybrid electric vehicle controller design problem.