Fast convergence of a primal-dual dynamical system with implicit Hessian damping and Tikhonov regularization

Hong-lu Li, Xin He, Yi-bin Xiao

Published 2025-06-24Version 1

This paper proposes two primal-dual dynamical systems for solving linear equality constrained convex optimization problems: one with implicit Hessian damping only, and the other further incorporating Tikhonov regularization. We analyze the fast convergence properties of both dynamical systems and show that they achieve the same convergence rates. To the best of our knowledge, this work provides the first theoretical analysis establishing a convergence rate $o(\frac{1}{t^2})$ for the primal-dual gap and a convergence rate $o(\frac{1}{t})$ for the velocity $\dot{x}(t)$, without imposing additional assumptions on the objective function beyond convexity and L-smoothness. Moreover, we show that the trajectory generated by the dynamical system with Tikhonov regularization converges strongly to the minimum-norm solution of the underlying problem. Finally, numerical experiments are conducted to validate the theoretical findings. Interestingly, the trajectories exhibit smooth behavior even when the objective function is only continuously differentiable.