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 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.
Related
- Strong convergence of a slowly damped primal-dual dynamical system with general time-scaling, Tikhonov regularization and implicit Hessian damping
- Lyapunov-Schmidt reduction for periodic orbits of implicit discrete dynamical systems
- An H-convergence-based implicit function theorem for homogenization of nonlinear non-smooth elliptic systems
- Solving the Tikhonov Regularization of the Total Least Squares Problem by matrix simultaneous diagonalization via congruence
- Inverse eigenvalue problem for discrete three-diagonal Schroedinger operator and the continuum limit in Tikhonov regularization approach
BibTeX
@misc{li2026convergence,
title = {{Convergence rates of mixed primal-dual dynamical systems with implicit Hessian damping and Tikhonov regularization}},
author = {Hong-lu Li and Yi-bin Xiao and Xin He and Kequan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39144},
eprint = {2610.39144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39144},
}
RIS
TY - UNPB
TI - Convergence rates of mixed primal-dual dynamical systems with implicit Hessian damping and Tikhonov regularization
AU - Hong-lu Li
AU - Yi-bin Xiao
AU - Xin He
AU - Kequan Zhao
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39144
UR - https://arxiv.org/abs/2610.39144
AB - 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/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.
ER -