Non-probabilistic set-based selection strategy for multi-objective optimization with interval uncertainties

Uncertainty introduces great challenges for multi-objective optimization, as the solutions are no longer deterministic values, complicating both the accurate evaluation of solution quality and the selection of elite solutions. Misjudging the dominance relationship among uncertain solutions may result in the loss of superior solutions, and imprecise crowding distance quantification may fail to maintain population diversity. Therefore, a novel non-probabilistic set-based selection strategy (NPS) is developed to balance the convergence and diversity of uncertain populations. It employs a novel two-dimensional interval dominance relationship and an interval crowding distance model to determine the new parent population. Additionally, a dimension-wise approach (DWA), a non-intrusive uncertainty analysis model, is used to quantify the bounds of uncertain optimization objectives. Furthermore, a novel interval crowding distance-based sample standard deviation metric is proposed to enhance the accuracy of diversity evaluation for uncertain populations. The proposed NPS is integrated into two classical multi-objective optimization frameworks and is compared with other selection strategies across multiple groups of benchmarks. The results indicate that algorithms incorporating NPS and DWA can not only effectively explore the Pareto Front under uncertainties but also directly evaluate uncertain solutions with limited samples. Compared with other selection strategies, NPS can explore an optimal solution set with superior convergence, higher diversity, and lower uncertainty.