Tikhonov regularized second-order plus first-order primal-dual dynamical systems with asymptotically vanishing damping for linear equality constrained convex optimization problems

TL;DR

This paper introduces a Tikhonov regularized primal-dual dynamical system with vanishing damping for solving linear equality constrained convex problems, demonstrating convergence rates and minimal norm solution attainment.

Contribution

It develops a novel dynamical system framework with asymptotically vanishing damping and analyzes its convergence properties based on the regularization parameter's decay rate.

Findings

Fast convergence rates when the regularization parameter decreases rapidly.

Strong convergence to the minimal norm solution with slow decay of the regularization parameter.

Numerical experiments confirm the method's efficiency.

Abstract

In this paper, in the setting of Hilbert spaces, we consider a Tikhonov regularized second-order plus first-order primal-dual dynamical system with asymptotically vanishing damping for a linear equality constrained convex optimization problem. The convergence properties of the proposed dynamical system depend heavily upon the choice of the Tikhonov regularization parameter. When the Tikhonov regularization parameter decreases rapidly to zero, we establish the fast convergence rates of the primal-dual gap, the objective function error, the feasibility measure, and the gradient norm of the objective function along the trajectory generated by the system. When the Tikhonov regularization parameter tends slowly to zero, we prove that the primal trajectory of the Tikhonov regularized dynamical system converges strongly to the minimal norm solution of the linear equality constrained convex optimization problem. Numerical experiments are performed to illustrate the efficiency of our approach.