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    Area of Science:

    • Optimization Theory
    • Control Systems Engineering
    • Distributed Computing

    Background:

    • Resource allocation is critical in various systems.
    • Existing methods struggle with local convex set constraints.
    • Nonsmoothness in optimization problems poses challenges.

    Purpose of the Study:

    • To design a distributed adaptive continuous-time optimization algorithm.
    • To address resource allocation problems with resource and local convex set constraints.
    • To overcome limitations of projection operator methods.

    Main Methods:

    • Laplacian-gradient method and adaptive control.
    • Distance-based exact penalty function for constraint handling.
    • Nonsmooth analysis and set-valued LaSalle invariance principle.

    Main Results:

    • The proposed algorithm effectively solves nonsmooth resource allocation problems.
    • The penalty function method successfully handles local convex sets.
    • Theoretical results are validated through simulation examples.

    Conclusions:

    • The developed algorithm offers a robust solution for constrained resource allocation.
    • The novel approach advances distributed optimization techniques.
    • This work provides a foundation for future research in adaptive control and optimization.