A Unified Primal and Dual Consensus Algorithm With Predefined-Time Convergence for Time-Varying Optimization

This article proposes a unified consensus algorithm with predefined-time convergence aimed at solving time-varying optimization problems based on multiagent systems under connected and undirected networks. The goal is to minimize the sum of local objective functions in a predefined time, where each agent possesses knowledge only of its local objective function. Two problems are addressed in this article: 1) optimal consensus and 2) optimal resource allocation, in which both the local objective function and the coupled equality constraint vary with time. The proposed unified consensus algorithm achieves the predefined-time optimization by leveraging the predefined-time stability theory under mild assumptions. Specifically, by transforming into the Lagrange dual problem, the optimal resource allocation problem is solved by the unified consensus algorithm from the perspective of dual variables. It is worth mentioning that the proposed unified consensus algorithm does not rely on the second-order information of objective functions, including the partial derivative of the gradient concerning time and the Hessian information. Finally, two simulation examples compared with state-of-the-art methods, application to the large-scale system, and application to the target encirclement problem of multirobot systems are provided to substantiate the theoretical results.