Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
Abstract
In this paper, we study a Tikhonov-regularized mixed-order primal--dual dynamical system with implicit Hessian damping for linearly constrained convex optimization problems in finite-dimensional Euclidean spaces, where the primal equation is second order and incorporates the viscous damping term δ√(ε(t)) ẋ(t), whereas the multiplier equation remains first order. By constructing a new class of energy functions, for a general Tikhonov regularization coefficient ε(t), we prove the strong convergence of the primal trajectory and derive fast convergence rates under the same parameter assumptions, without imposing any eventual inside/outside-ball condition. More precisely, the primal trajectory converges to the minimum-norm solution, and the multiplier converges to a compatible KKT multiplier, while the convergence rates of the Lagrangian gap, feasibility violation, and objective residual are o(ε(t)), and the convergence rate of the velocity norm is o(√(ε(t))). For the critical case ε(t)=c/t2, in which the damping coefficient δ√(ε(t)) reduces to δ√c/t, we establish the sharper convergence rates o(t−2) for the Lagrangian gap, feasibility violation, and objective residual, together with o(t−1) for the velocity norm, which improve the corresponding O(t−2) and O(t−1) decay estimates obtained in the related literature. Most importantly, when the proposed dynamical system is specialized to the finite-dimensional unconstrained setting, our analysis answers the open question on strong convergence in this critical regime posed by Attouch and László [Math. Methods Oper. Res., 99 (2024), pp.~307--347].
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BibTeX
@misc{li2026fast,
title = {{Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions}},
author = {Hong-lu Li and Yi-bin Xiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06420},
eprint = {2609.06420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06420},
}
RIS
TY - UNPB
TI - Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
AU - Hong-lu Li
AU - Yi-bin Xiao
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06420
UR - https://arxiv.org/abs/2609.06420
AB - In this paper, we study a Tikhonov-regularized mixed-order primal--dual dynamical system with implicit Hessian damping for linearly constrained convex optimization problems in finite-dimensional Euclidean spaces, where the primal equation is second order and incorporates the viscous damping term δ√(ε(t)) ẋ(t), whereas the multiplier equation remains first order. By constructing a new class of energy functions, for a general Tikhonov regularization coefficient ε(t), we prove the strong convergence of the primal trajectory and derive fast convergence rates under the same parameter assumptions, without imposing any eventual inside/outside-ball condition. More precisely, the primal trajectory converges to the minimum-norm solution, and the multiplier converges to a compatible KKT multiplier, while the convergence rates of the Lagrangian gap, feasibility violation, and objective residual are o(ε(t)), and the convergence rate of the velocity norm is o(√(ε(t))). For the critical case ε(t)=c/t^2, in which the damping coefficient δ√(ε(t)) reduces to δ√c/t, we establish the sharper convergence rates o(t^−2) for the Lagrangian gap, feasibility violation, and objective residual, together with o(t^−1) for the velocity norm, which improve the corresponding O(t^−2) and O(t^−1) decay estimates obtained in the related literature. Most importantly, when the proposed dynamical system is specialized to the finite-dimensional unconstrained setting, our analysis answers the open question on strong convergence in this critical regime posed by Attouch and László [Math. Methods Oper. Res., 99 (2024), pp.~307--347].
ER -