Global and exponential convergence of a primal-dual dynamical system approach for the separable convex optimization problem

In this paper, we propose a primal-dual dynamical system for the separable convex optimization problem min x,y f (x) + g(y), s.t.Ax + By = b.Under the convexity assumptions of f and g, we show that the trajectory of the proposed primal-dual dynamical system globally converges to a saddle point of the problem as the time t → +∞.When f and/or g satisfy strong convexity and Lipschitz continuity assumptions, along with full rank assumptions on A and/or B, we prove the exponential convergence of the dynamical system approach.Besides, under the metric subregularity condition, we show that the exponential convergence can also be guaranteed without strong convexity and full rank assumptions.Finally, we give numerical results to show the practical performance of the dynamical system approach.