Tikhonov-Regularized Second-Order Primal-Dual Dynamics for Convex-Concave Bilinear Saddle Point Problems

Abstract

This paper develops a Lyapunov-based coefficient-compatible framework for a class of second-order primal-dual dynamical systems with Tikhonov regularization. Unlike existing accelerated inertial primal-dual dynamics, whose analyses are typically carried out under prescribed damping profiles, the proposed framework characterizes the interplay among the viscous damping coefficient, the time-scaling coefficient, the linear extrapolation parameter, and the Tikhonov regularization parameter through differential compatibility conditions, thereby accommodating time-varying coefficients beyond prescribed damping profiles. Within this framework, we investigate two cases in which the Tikhonov regularization parameter decays to zero at different rates. When the Tikhonov regularization parameter decays to zero relatively rapidly, the fast convergence rate of the primal-dual gap is preserved, whereas a slower decay rate yields refined asymptotic convergence estimates. We further show that a sufficiently slow decay ensures strong convergence of the trajectories to the minimum-norm saddle point. To establish the latter property, we introduce a moving-saddle Lyapunov framework centered at the instantaneous regularized saddle point and derive verifiable coefficient conditions ensuring the required tracking behavior. Finally, three numerical examples are presented to illustrate the theoretical results.