Prescribed-time fully distributed optimization for time-varying costs: Zero-gradient-sum scheme.

This article introduces a novel distributed optimization algorithm for time-varying objectives that achieves prescribed-time convergence. The method is developed based on zero-gradient-sum (ZGS) principles and employs a sliding-mode control framework. Specifically, the controller is designed to satisfy the ZGS condition within a prescribed time horizon, ensuring exact convergence by the user-specified deadline. A key contribution of this research is the introduction of adaptive parameters with time-varying scaling functions. This innovation addresses a fundamental limitation in existing methods by eliminating their dependency on global network information. Consequently, the algorithm achieves truly distributed control without requiring knowledge of Laplacian matrix eigenvalues. The algorithm offers significant advantages including complete freedom from initial condition constraints and local minimization requirements, full independence from global topological information, and rigorous prescribed-time convergence guarantees. Theoretical analysis establishes the convergence properties through an integrated approach that combines optimization theory with Lyapunov stability analysis. Numerical simulations demonstrate the algorithm's effectiveness and superior performance in handling time-varying optimization problems compared to existing approaches.