Distributed Optimization in Open Networks under Redundancy

This paper addresses the problem of distributed optimization in an open network setting where new agents can join and existing ones can leave the network at specific times. Each agent has a local cost function, and the collective goal is to minimize the aggregate cost. As the number of agents is time-varying, the optimal solutions to this problem are time-dependent, making finding an exact solution generally challenging. In this paper, we will study this distributed optimization under a redundancy condition. Under this property, we will show that solving this time-varying optimization problem is essentially equivalent to solving a static optimization problem. Our main focus is to show that the classic distributed local gradient descent method in the client-server architecture will converge exactly to an optimal solution. In addition, we provide a formula to characterize the convergence properties of this method in the setting of open networks under the redundancy condition.