Asymptotic behavior of the Arrow-Hurwicz differential system with Tikhonov regularization

TL;DR

This paper studies the long-term behavior of solutions to the Arrow-Hurwicz differential system with Tikhonov regularization, showing convergence to the minimal norm solution and providing decay rate estimates under various conditions.

Contribution

It introduces new conditions on Tikhonov terms that ensure strong convergence and derives explicit decay rates for the solutions and gap functions.

Findings

Solutions converge strongly to the least norm element of zeros.

Decay rates for primal-dual gap and solution velocity are established.

Refined exponential decay estimates are provided for decreasing Tikhonov terms.

Abstract

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.