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

Abstract

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 $\mathcal{O}(t^{-2})$ 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 $\mathcal{O}(k^{-2})$ rates at the extrapolated primal iterates. In particular, in the critical regularization regimes $ε(t)=a/t^2$ and $ε_k=a/k^2$, 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.