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    Area of Science:

    • Control Theory
    • Optimization Algorithms
    • Distributed Systems

    Background:

    • Time-varying optimization problems present challenges in dynamic environments.
    • Existing algorithms may struggle to adapt to continuously changing cost and demand functions.
    • Resource allocation problems often exhibit time-varying characteristics.

    Purpose of the Study:

    • To design novel distributed algorithms for time-varying optimization problems.
    • To develop algorithms capable of handling both unconstrained and constrained (resource allocation) scenarios.
    • To ensure convergence of solutions to time-varying optimal states.

    Main Methods:

    • Development of new distributed algorithm structures.
    • Design of algorithms for distributed time-varying optimization (DTVO) and distributed time-varying optimal resource allocation (DTVORA).
    • Utilizing exosystem dynamics for driving terms to track solution variations.
    • Convergence analysis using Lyapunov theory.

    Main Results:

    • Demonstrated existence of time-varying solutions for the addressed problems.
    • Ensured convergence of solution states to these time-varying solutions.
    • Validation of algorithm performance through illustrative examples.

    Conclusions:

    • The proposed distributed algorithms effectively address time-varying optimization and resource allocation.
    • Exosystem dynamics provide a robust mechanism for tracking dynamic changes.
    • The algorithms show potential for real-world applications requiring adaptive optimization.