A Continuous-Time Nesterov Accelerated Gradient Method for Distributed Convex Optimization with an Exponential Convergent Bregman Distance

In this work, we propose a continuous-time second-order accelerated gradient method for distributed optimization problems. It is proven that the proposed proportional-integral and proportional-only algorithms guarantee that the generalized Bregman distance (Note: For a continuously differentiable convex function f(x), the Bregman distance associated with two points x,y ∈ℝ
ⁿ
is a distance equal to f(x)− f(y)−∇f(y)
^T
(x − y). The generalized Bregman distance refers to its generalization to nonsmooth functions) associated with the agents’ states and an optimal solution could be exponentially convergent, provided that the local objective functions are convex and there exists at least one optimal solution for the optimization problem. Furthermore, we show that the proposed method achieves asymptotic convergence for strictly convex functions and exponential convergence for strong convex functions as are commonly achieved by most of the existing algorithms. The robustness of the algorithm is further discussed.