A General Tikhonov Regularized Second-Order Dynamical System for Convex-Concave Bilinear Saddle Point Problems

TL;DR

This paper introduces a Tikhonov regularized second-order dynamical system for convex-concave bilinear saddle point problems, analyzing its convergence properties and demonstrating its effectiveness through numerical experiments.

Contribution

It develops a new dynamical system with Tikhonov regularization for saddle point problems, establishing convergence rates and strong convergence to minimum-norm solutions.

Findings

Convergence rate is O(1/t^2 * beta(t)) with rapid regularization decay.

Convergence rate is o(1/beta(t)) with slow regularization decay.

Numerical experiments confirm the theoretical convergence properties.

Abstract

In this paper, we propose a general Tikhonov regularized second-order dynamical system with viscous damping, time scaling and extrapolation coefficients for the convex-concave bilinear saddle point problem. By the Lyapunov function approach, we show that the convergence properties of the proposed dynamical system depend on the choice of the Tikhonov regularization parameter. Specifically, when the Tikhonov regularization parameter tends to zero rapidly, the convergence rate of the primal-dual gap along the generated trajectory is O(1 over t squared times beta(t)); when the Tikhonov regularization parameter tends to zero slowly, the convergence rate of the primal-dual gap is o(1 over beta(t)). We also prove the strong convergence property of the trajectory generated by the Tikhonov regularized dynamical system to the minimum-norm solution of the convex-concave bilinear saddle point problem, and derive several integral estimates. In addition, the effectiveness of the proposed dynamical system is verified through a series of numerical experiments.