Distributed continuous-time online optimization using saddle-point methods
This paper introduces saddle-point methods for distributed continuous-time online convex optimization, where the system objective function varies arbitrarily over time subject to some global inequality constraints. The overall dynamics of the proposed saddle-point controller are described by a system of differential equations, coupled linearly through the network Laplacian. The controller pushes actions along the negative gradient direction of the objective, constraint violation, as well as network disagreement using only causal and locally available information, while dynamically adapting the Lagrange multipliers in a decentralized fashion. We define regret as the cost difference with the optimal action over time. We show that the proposed saddle-point controller achieves a regret of order O(√T) with the time horizon T. We also address the impact of the network topology, encoded in the spectrum of the network Laplacian, as a factor on the speed of convergence.
