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    Area of Science:

    • Computational Mathematics
    • Optimization Algorithms
    • Artificial Intelligence

    Background:

    • Evolutionary algorithms have been widely used for solving nonlinear equation systems (NESs) for two decades.
    • Existing benchmark test sets for NESs often lack crucial features like high dimensionality and large search ranges, limiting their real-world applicability.
    • This gap hinders the effective evaluation and development of algorithms for complex, real-world problems.

    Purpose of the Study:

    • To address the limitations of current benchmark test sets for nonlinear equation systems.
    • To propose a general toolkit for generating artificial test problems with diverse and realistic characteristics.
    • To introduce and evaluate a novel method for solving these challenging NESs.

    Main Methods:

    • Development of a general toolkit for generating artificial nonlinear equation systems (NESs).
    • Construction of 18 diverse test instances designed to represent real-world problem complexities.
    • Introduction of a transformation method to convert NESs into single-objective optimization problems.
    • Design of a two-phase method to solve the transformed multimodal optimization problems.

    Main Results:

    • The proposed benchmark test set reveals that current algorithms exhibit poor performance on challenging NESs.
    • The newly developed two-phase method demonstrates superior or competitive performance compared to existing algorithms.
    • The generated test instances effectively capture critical features of real-world problems, such as high dimensionality and large search spaces.

    Conclusions:

    • The developed benchmark and toolkit represent a significant advancement in evaluating algorithms for nonlinear equation systems.
    • The proposed two-phase method offers an effective approach for tackling complex, multimodal optimization problems derived from NESs.
    • Further research into realistic benchmark design is crucial for advancing the field of evolutionary computation for NESs.