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    This study introduces a novel method for bilevel optimization, significantly reducing computational cost by estimating unknown mappings. The approach accelerates convergence towards optimal solutions, outperforming existing methods.

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    Area of Science:

    • Optimization Theory
    • Computational Mathematics
    • Applied Mathematics

    Background:

    • Bilevel optimization problems are computationally expensive due to nested structures requiring numerous inner problem solutions.
    • Existing methods for reducing bilevel problems often assume known forms (e.g., piecewise linear, convex) for reaction set or lower-level optimal value function mappings.
    • Estimating these mappings is crucial but challenging when their forms are unknown.

    Purpose of the Study:

    • To develop a general bilevel optimization method that does not require prior assumptions on mapping structures.
    • To leverage both reaction set and lower-level optimal value function mappings simultaneously for enhanced problem reduction.
    • To significantly reduce the computational burden and function evaluations in solving bilevel optimization problems.

    Main Methods:

    • Utilizes Kriging approximations within an evolutionary algorithm, employing population members as samples for metamodel creation.
    • Develops an auxiliary optimization problem using a Kriging-based metamodel of the lower-level optimal value function.
    • Employs the auxiliary problem within a local search strategy to accelerate convergence towards bilevel optimal solutions.

    Main Results:

    • The proposed Kriging-based evolutionary algorithm effectively solves general bilevel optimization problems without structural assumptions on mappings.
    • Experimental results on test problems and a control theory application demonstrate significant savings in lower-level function evaluations.
    • The approach outperforms state-of-the-art methods for bilevel optimization, showing substantial computational efficiency gains.

    Conclusions:

    • The combined estimation of both mappings, facilitated by Kriging approximations and an auxiliary problem, offers a powerful approach for general bilevel optimization.
    • This method provides substantial computational savings, particularly in the number of function evaluations for the lower-level problem.
    • The proposed technique is a promising advancement for tackling complex bilevel optimization challenges across various domains.