Journal of Convex Analysis 33 (2026), No. 3&4, 565--596
Copyright Heldermann Verlag 2026
Asymptotic Behavior of the Arrow-Hurwicz Differential System with Tikhonov Regularization
Fouad Battahi
Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco
f.battahi.ced@uca.ac.ma
Zaki Chbani
Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco
chbaniz@uca.ac.ma
Simon K. Niederländer
University of Applied Sciences, Ingolstadt, Germany
simon.niederlaender@thi.de
Hassan Riahi
Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco
h-riahi@uca.ac.ma
In a real Hilbert space setting, we investigate the asymptotic behavior of the solutions of the classical Arrow-Hurwicz differential system combined with Tikhonov regularizing terms. Under some newly proposed conditions on the Tikhonov terms involved, we show that the solutions of the regularized Arrow-Hurwicz differential system strongly converge toward the element of least norm within its set of zeros. Moreover, we provide fast asymptotic decay rate estimates for the so-called primal-dual gap function and the norm of the solutions' velocity. If, in addition, the Tikhonov regularizing terms are decreasing, we provide some refined estimates in the sense of an exponentially weighted moving average. Under the additional assumption that the governing operator of the Arrow-Hurwicz differential system satisfies a reverse Lipschitz condition, we further provide a fast rate of strong convergence of the solutions toward the unique zero. We conclude our study by deriving the corresponding decay rate estimates with respect to the so-called viscosity curve. Numerical experiments illustrate our theoretical findings.
Keywords: Arrow-Hurwicz differential system, Tikhonov regularization, viscosity curve, convex minimization, saddle-value problem.
MSC: 37N40, 46N10; 49K35, 90C25.
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