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    This study introduces novel algorithms for distributed optimization problems with partial-impact cost functions. The research establishes a link between algorithm equilibrium and optimization, demonstrating efficiency through numerical examples.

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    Area of Science:

    • Optimization Theory
    • Distributed Systems
    • Nonsmooth Analysis

    Background:

    • Distributed optimization problems are prevalent in various fields.
    • Partial-impact cost functions introduce complexity in optimization.
    • Existing methods may not efficiently handle these specific problem structures.

    Purpose of the Study:

    • To address a specific distributed optimization problem involving partial-impact cost functions.
    • To develop and present two novel algorithms for solving this problem: a structured and a gradient-based approach.
    • To establish a theoretical connection between the equilibrium of the proposed algorithms and the optimization problem itself.

    Main Methods:

    • Development of two distinct algorithms: one structured and one gradient-based.
    • Application of nonsmooth analysis tools.
    • Utilizing the change of coordinate theorem.
    • Validation through two numerical examples with practical relevance.

    Main Results:

    • Successful design and presentation of two algorithms for the target optimization problem.
    • Establishment of a theoretical link between algorithm equilibrium and the optimization problem.
    • Demonstration of the designed algorithms' efficiency via numerical simulations.

    Conclusions:

    • The proposed algorithms offer effective solutions for distributed optimization with partial-impact cost functions.
    • The theoretical framework provides a deeper understanding of the relationship between algorithm dynamics and optimization outcomes.
    • Numerical results confirm the practical applicability and efficiency of the developed methods.