Fast convergence of dynamical systems with implicit Hessian damping and Tikhonov regularization

TL;DR

This paper introduces new primal-dual dynamical systems with implicit Hessian damping and Tikhonov regularization for convex optimization, achieving fast convergence and strong convergence to minimum norm solutions, validated by numerical experiments.

Contribution

It presents a novel primal-dual dynamical system with implicit Hessian damping and Tikhonov regularization, enhancing convergence properties without Hessian computation.

Findings

The proposed system achieves fast convergence under certain conditions.

Inclusion of Tikhonov regularization ensures convergence to minimum norm solutions.

Numerical experiments confirm theoretical convergence results.

Abstract

This paper proposes novel primal-dual dynamical systems for solving linear equality constrained convex optimization. First, we introduce a primal-dual dynamical system with implicit Hessian damping, which can neutralize the transversal oscillations without requiring computation of the Hessian matrix. We establish the fast convergence properties of the proposed dynamical system under suitable conditions. Furthermore, we incorporate a Tikhonov regularization term and prove that the resulting trajectories converge strongly to the minimum norm solution. Numerical experiments are conducted to validate the theoretical findings.