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    Area of Science:

    • Optimization Theory
    • Distributed Systems
    • Convex Analysis

    Background:

    • Constrained optimization problems are fundamental in various scientific and engineering fields.
    • Solving these problems in a distributed manner is crucial for large-scale systems.
    • Existing methods often struggle with distributed constraints and gradient tracking.

    Purpose of the Study:

    • To develop distributed optimization algorithms for problems with convex cost functions and convex set constraints.
    • To enable decentralized computation using only local information exchange.
    • To address the limitations of classical projection methods in gradient-tracking frameworks.

    Main Methods:

    • Design of two gradient-tracking-based distributed optimization algorithms.
    • Employment of a novel indirect projection method for closed convex set constraints.
    • Introduction of two-time scale analysis for convergence.
    • Analysis over both weight-balanced and weight-unbalanced graphs.

    Main Results:

    • The proposed algorithms effectively solve the constrained optimization problem in a distributed manner.
    • Demonstration of the efficacy of the indirect projection method for handling constraints.
    • Proof of linear convergence rates for the algorithms under strong convexity and L-smoothness conditions with fixed step sizes.

    Conclusions:

    • The developed algorithms offer a viable solution for distributed constrained optimization.
    • The indirect projection method is a key innovation for handling constraints within gradient-tracking.
    • The study provides theoretical guarantees for the performance of the distributed algorithms.