Strong convergence of a slowly damped primal-dual dynamical system with general time-scaling, Tikhonov regularization and implicit Hessian damping

Abstract

This paper proposes a novel mixed-order inertial primal-dual dynamical system with general time-scaling and Tikhonov regularization coefficients, together with implicit Hessian damping, for solving convex optimization problems with linear equality constraints in Hilbert spaces. The proposed dynamical system incorporates slowly vanishing viscous damping and is developed based on a time-dependent Tikhonov-regularized Lagrangian. Under suitable assumptions, we establish fast convergence rates for the objective residual, the feasibility violation, and the primal-dual gap using a Lyapunov analysis. Moreover, we prove that the generated primal-dual trajectory converges strongly to the minimum-norm primal-dual solution. Numerical experiments illustrate the theoretical results and indicate improved convergence performance relative to the comparison dynamical systems.