Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems
Abstract
In this paper, we study a Tikhonov-regularized second-order primal--dual dynamical system for smooth nonlinear convex--concave saddle-point problems in real Hilbert spaces. The dynamical system incorporates a vanishing damping coefficient α/t, a Hessian-driven term, and a critical Tikhonov regularization coefficient c/t2. By constructing suitable Lyapunov functions, we establish strong convergence to the minimum-norm saddle point and fast convergence rates under the same parameter conditions, without imposing any eventual inside/outside-ball condition. Specifically, the primal--dual trajectory converges strongly to the minimum-norm saddle point, while the primal--dual gap and the velocity norm decay at rates o(t−2) and o(t−1), respectively. Furthermore, we propose an inexact primal--dual algorithm with a computable relative residual criterion and prove strong convergence of the generated sequence to the same minimum-norm saddle point, together with an o(k−2) convergence rate for the primal--dual gap. Finally, numerical experiments on nonlinear and bilinear problems of different dimensions illustrate the minimum-norm selection property and compare the performance of the proposed dynamical system and algorithm with related methods.
Related
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BibTeX
@misc{li2026tikhonovregularized,
title = {{Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems}},
author = {Hong-lu Li and Yi-bin Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39133},
eprint = {2610.39133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39133},
}
RIS
TY - UNPB
TI - Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems
AU - Hong-lu Li
AU - Yi-bin Xiao
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39133
UR - https://arxiv.org/abs/2610.39133
AB - In this paper, we study a Tikhonov-regularized second-order primal--dual dynamical system for smooth nonlinear convex--concave saddle-point problems in real Hilbert spaces. The dynamical system incorporates a vanishing damping coefficient α/t, a Hessian-driven term, and a critical Tikhonov regularization coefficient c/t^2. By constructing suitable Lyapunov functions, we establish strong convergence to the minimum-norm saddle point and fast convergence rates under the same parameter conditions, without imposing any eventual inside/outside-ball condition. Specifically, the primal--dual trajectory converges strongly to the minimum-norm saddle point, while the primal--dual gap and the velocity norm decay at rates o(t^−2) and o(t^−1), respectively. Furthermore, we propose an inexact primal--dual algorithm with a computable relative residual criterion and prove strong convergence of the generated sequence to the same minimum-norm saddle point, together with an o(k^−2) convergence rate for the primal--dual gap. Finally, numerical experiments on nonlinear and bilinear problems of different dimensions illustrate the minimum-norm selection property and compare the performance of the proposed dynamical system and algorithm with related methods.
ER -