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    This study introduces a decentralized-partial-consensus optimization algorithm using recurrent neural networks to solve problems with inequality constraints. The method ensures convergence, demonstrating effectiveness through examples.

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    Area of Science:

    • Control Systems
    • Optimization Theory
    • Networked Systems

    Background:

    • Decentralized optimization problems are prevalent in networked systems.
    • Handling inequality constraints in these systems presents significant challenges.
    • Existing methods may lack efficiency or scalability for partial consensus.

    Purpose of the Study:

    • To propose a novel decentralized-partial-consensus optimization (DPCO) framework.
    • To address optimization problems incorporating inequality constraints.
    • To develop a robust algorithm for achieving partial consensus in dynamic networks.

    Main Methods:

    • Construction of a partial-consensus matrix derived from graph Laplacian.
    • Development of a continuous-time algorithm utilizing interconnected recurrent neural networks (RNNs).
    • Application of nonsmooth analysis and Lyapunov theory for convergence proofs.

    Main Results:

    • A novel DPCO problem formulation is presented.
    • A continuous-time RNN-based algorithm is derived to solve the DPCO problem.
    • Theoretical convergence guarantees for the algorithm are established using advanced mathematical tools.

    Conclusions:

    • The proposed DPCO framework effectively handles inequality constraints in decentralized settings.
    • The RNN-based algorithm demonstrates reliable convergence properties.
    • Numerical examples validate the practical applicability and efficiency of the developed method.