Inertial primal-dual dynamics with Hessian-driven damping and Tikhonov regularization for convex-concave bilinear saddle point problems

TL;DR

This paper introduces a second-order primal-dual dynamical system with Hessian-driven damping and Tikhonov regularization, achieving fast convergence and strong trajectory convergence for convex-concave saddle point problems.

Contribution

It develops a novel dynamical system that combines Hessian-driven damping and Tikhonov regularization, ensuring improved convergence rates and trajectory stability.

Findings

Fast convergence rate of primal-dual gap along trajectories

Simultaneous convergence rate and strong convergence achieved

Numerical examples demonstrate system effectiveness

Abstract

This paper deals with a second-order primal-dual dynamical system with Hessian-driven damping and Tikhonov regularization terms in connection with a convex-concave bilinear saddle point problem. We first obtain a fast convergence rate of the primal-dual gap along the trajectory generated by the dynamical system, and provide some integral estimates. Then, based on the setting of the parameters involved, we demonstrate that both the convergence rate of the primal-dual gap and the strong convergence of the trajectory can be achieved simultaneously. Furthermore, we evaluate the performance of the proposed system using two numerical examples.