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    A new Trust-Tech source-point method systematically finds multiple local optimal solutions (LOSs) for nonlinear optimization problems. This approach successfully identified thousands of LOSs and global optima for high-dimensional test functions.

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    Area of Science:

    • Optimization
    • Computational Mathematics
    • Numerical Analysis

    Background:

    • Continuous unconstrained nonlinear optimization problems often possess numerous local optimal solutions (LOSs).
    • Existing methods may struggle to systematically identify multiple LOSs or escape from a found solution.
    • Finding multiple LOSs is crucial for comprehensively understanding the solution landscape.

    Purpose of the Study:

    • To propose a novel Trust-Tech source-point method for systematically computing multiple local optimal solutions (LOSs).
    • To develop a theoretical foundation for escaping LOSs and entering new stability regions.
    • To numerically validate the method's efficacy in finding numerous LOSs and global optima.

    Main Methods:

    • The proposed Trust-Tech source-point method involves four stages: finding an initial LOS, escaping it, entering a new stability region (SR), and computing other stable equilibrium points (SEPs) or LOSs.
    • Theoretical underpinnings for the escape and entry stages were developed.
    • Numerical evaluations were performed on high-dimensional test functions.

    Main Results:

    • The method successfully computed a large number of LOSs, for example, 5085 LOSs on a 50-dimensional test function.
    • The Trust-Tech source-point method demonstrated the capability to find global optimal solutions for several 50-D and 100-D test functions.
    • The theoretical framework for escaping and entering SRs proved to be general.

    Conclusions:

    • The proposed Trust-Tech source-point method is an effective approach for systematically computing multiple local optimal solutions in continuous unconstrained nonlinear optimization.
    • The method offers a robust way to explore the solution space and identify global optima.
    • The developed theoretical foundation enhances the understanding of solution landscape navigation in optimization.