Distributed Finite-Time Optimization

In this paper, we consider the distributed optimization problem of multi-agent systems. The objective is to minimize the global objective function, which is the sum of local objective functions, by using local communication and local computation. We develop a distributed proportional-integral (PI) algorithm, based on the information received from its neighboring agents through the communication network and the gradient of its own objective function. For the case of quadratic objective functions, we establish sufficient conditions on the gain parameters under which the algorithm exponentially converges to the unique global minimizer. Moreover, we equip the proposed algorithm with a decentralized algorithm, which enables an arbitrarily chosen agent to compute the exact global minimizer within a finite number of time steps, using its own states observed over a successive time steps. Finally, the theoretical results are illustrated by numerical simulations.