Tikhonov regularized inertial primal-dual dynamics for convex-concave bilinear saddle point problems

TL;DR

This paper introduces a Tikhonov regularized inertial primal-dual dynamical system for convex-concave saddle point problems, analyzing its convergence behavior under different regularization decay rates and supporting findings with numerical experiments.

Contribution

It proposes a novel second-order primal-dual dynamical system with Tikhonov regularization, providing convergence analysis under various decay conditions of the regularization parameter.

Findings

Convergence rates for primal-dual gap and velocity vector when regularization decreases rapidly.

Strong convergence to minimal norm solution when regularization decreases slowly.

Numerical experiments validating theoretical convergence results.

Abstract

In this paper, for a convex-concave bilinear saddle point problem, we propose a Tikhonov regularized second-order primal-dual dynamical system with slow damping, extrapolation and general time scaling parameters. Depending on the vanishing speed of the rescaled regularization parameter (i.e., the product of Tikhonov regularization parameter and general time scaling parameter), we analyze the convergence properties of the trajectory generated by the dynamical system. When the rescaled regularization parameter decreases rapidly to zero, we obtain convergence rates of the primal-dual gap and velocity vector along the trajectory generated by the dynamical system. In the case that the rescaled regularization parameter tends slowly to zero, we show the strong convergence of the trajectory towards the minimal norm solution of the convex-concave bilinear saddle point problem. Further, we also present some numerical experiments to illustrate the theoretical results.