Time-varying distributed optimization for a class of stochastic multi-agent systems

Distributed optimization problems have received much attention due to their privacy preservation, parallel computation, less communication, and strong robustness. This paper presents and studies the time-varying optimization problems for a class of stochastic multi-agent systems for the first time. We first design a centralized protocol that ensures that the agent's tracking error on the optimal trajectory is exponentially ultimately bounded in a mean-square sense via stochastic Lyapunov theory. We then extend this approach to the distributed case. Therein, we propose a fixed-time estimator which guarantees that the global variables are estimated within a fixed time. Subsequently, based on this estimator, we develop a novel distributed protocol. Theoretical analysis again utilizes stochastic Lyapunov techniques to confirm that the tracking errors of all agents remain exponentially ultimately bounded in a mean-square sense. Finally, we validate our theoretical findings with numerical simulations that demonstrate the effectiveness of the protocol.