Tikhonov regularization of second-order plus first-order primal-dual dynamical systems for separable convex optimization

TL;DR

This paper introduces a Tikhonov regularized primal-dual dynamical system combining second-order and first-order differential equations, demonstrating improved convergence properties for solving separable convex optimization problems with linear constraints.

Contribution

It presents a novel Tikhonov regularized second-order plus first-order primal-dual dynamical system with time scaling, providing new convergence analysis and numerical validation.

Findings

Convergence of primal-dual gap and objective error established

Strong convergence to minimal norm solution proven

Numerical experiments show improved convergence rates

Abstract

This paper deals with a Tikhonov regularized second-order plus first-order primal-dual dynamical system with time scaling for separable convex optimization problems with linear equality constraints. This system consists of two second-order ordinary differential equations for the primal variables and one first-order ordinary differential equation for the dual variable.By utilizing the Lyapunov analysis approach, we obtain the convergence properties of the primal-dual gap, the objective function error, the feasibility measure and the gradient norm of the objective function along the trajectory. We also establish the strong convergence of the primal trajectory generated by the dynamical system towards the minimal norm solution of the separable convex optimization problem. Furthermore, we give numerical experiments to illustrate the theoretical results, showing that our dynamical system performs better than those in the literature in terms of convergence rates.