Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems

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.