A Decomposition Method for Both Additively and Nonadditively Separable Problems

Problem decomposition is crucial for coping with large-scale global optimization problems, which relies heavily on highly precise variable grouping methods. The state-of-the-art decomposition methods identify separability based on the finite differences principle, which is valid only for additively separable functions but not applicable to nonadditively separable functions. Therefore, we need to investigate separability in more depth in order to propose a more general principle and design more universal decomposition methods. In this article, we conduct a comprehensive theoretical investigation on separability, the core of which is proposing an innovative separability identification principle: the minimum points shift principle. By utilizing the new principle, we develop a general separability grouping (GSG) method that can handle both additively and nonadditively separable functions with high accuracy. In addition, we design a new set of benchmark functions based on nonadditive separability, which compensates for the lack of nonadditively separable functions in the previous test suites. Extensive experiments demonstrate that the proposed GSG achieves high grouping accuracy on both new and CEC series benchmark problems, especially on nonadditively separable problems Finally, we verify that the proposed GSG can effectively improve the optimization performance of nonadditively separable problems through optimization experiments.