Convergence Analysis of Hessian-Damped Tikhonov Regularized Dynamics with Oscillation Control for Convex-Concave Bilinear Saddle Point Problems

TL;DR

This paper introduces a second-order dynamical system with Tikhonov regularization and Hessian damping for convex-concave saddle point problems, analyzing its convergence and oscillation control.

Contribution

It develops a new dynamical system framework with time-varying parameters and proves strong convergence to minimum-norm solutions, highlighting Hessian damping's effectiveness.

Findings

Convergence rate of primal-dual gap established

Trajectories converge strongly to minimum-norm solutions

Hessian damping reduces oscillations effectively

Abstract

In this paper, we propose a class of general second-order primal-dual dynamical systems with Tikhonov regularization and Hessian-driven damping for solving convex-concave bilinear saddle point problems. The proposed dynamical system incorporates five general time-varying terms: viscous damping, time scaling, extrapolation, Tikhonov regularization, and Hessian-driven damping parameters. Under suitable parametric conditions, we analyze the asymptotic convergence properties of the dynamical system by constructing appropriate Lyapunov functions. Specifically, we obtain the convergence rate of the primal-dual gap and the boundedness of trajectories in the proposed dynamical system, and provide some integral estimates. Furthermore, we theoretically prove that the trajectories generated by the dynamical system converge strongly to the minimum-norm solution of the saddle point problem, and fully demonstrate that Hessian-driven damping can effectively alleviate oscillations. Finally, numerical experiments are conducted to verify the validity of the above theoretical results.