Convergence rates of mixed primal-dual dynamical systems with implicit Hessian damping and Tikhonov regularization

We propose a mixed-order primal--dual dynamical system with implicit Hessian damping and Tikhonov regularization for linearly constrained convex optimization, and derive a corresponding fast primal--dual algorithm. The proposed dynamical system couples a second-order primal equation with a first-order dual equation without requiring twice differentiability of the objective function. An energy analysis establishes O(t⁻²) convergence rates for the augmented Lagrangian gap, feasibility violation, and objective residual along an extrapolated primal trajectory. For the proposed algorithm, we construct discrete energy estimates and establish the corresponding O(k⁻²) rates at the extrapolated primal iterates. In particular, in the critical regularization regimes ε(t)=a/t² and εₖ=a/k², these accelerated rates hold simultaneously with strong convergence of the primal trajectory and primal iterates to the minimum-norm solution, respectively, under suitable parameter and trajectory conditions. Numerical experiments illustrate the convergence behavior of the proposed dynamical system and the corresponding algorithm.