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.

Author information

Contact details are reproduced from the original publication and may be historical.

Zaki Chbani

Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco

chbaniz@uca.ac.ma

Hassan Riahi

Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco

h-riahi@uca.ac.ma

Keywords

  • Arrow-Hurwicz differential system
  • Tikhonov regularization
  • viscosity curve
  • convex minimization
  • saddle-value problem

Mathematics Subject Classification

37N40, 46N10; 49K35, 90C25

References

  1. F. J. Aragón Artacho, M. H. Geoffroy: Characterization of metric regularity of subdifferentials, J. Convex Analysis 15 (2008) 365–380.https://doi.org/10.68381/jca15026Google Scholar
  2. K. J. Arrow, L. Hurwicz: A gradient method for approximating saddle points and constrained maxima, RAND Corporation, Santa Monica (1951), P–223, 15p.Google Scholar
  3. K. J. Arrow, L. Hurwicz, H. Uzawa: Studies in Linear and Non-Linear Programming, Stanford University Press, Stanford (1958).Google Scholar
  4. H. Attouch: Viscosity solutions of minimization problems, SIAM J. Optim. 6 (1996) 769–806.https://doi.org/10.1137/S1052623493259616Google Scholar
  5. H. Attouch, A. Balhag, Z. Chbani, H. Riahi: Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution, J. Diff. Equations 311 (2022) 29–58.https://doi.org/10.1016/j.jde.2021.12.005Google Scholar
  6. H. Attouch, A. Cabot, M.-O. Czarnecki: Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc. 370 (2018) 755–790.https://doi.org/10.1090/tran/6965Google Scholar
  7. H. Attouch, Z. Chbani, H. Riahi: Combining fast inertial dynamics for convex optimization with Tikhonov regularization, J. Math. Analysis Appl. 457 (2018) 1065–1094.https://doi.org/10.1016/j.jmaa.2016.12.017Google Scholar
  8. H. Attouch, R. Cominetti: A dynamical approach to convex minimization coupling approximation with the steepest descent method, J. Diff. Equations 128 (1996) 519–540.https://doi.org/10.1006/jdeq.1996.0104Google Scholar
  9. H. Attouch, M.-O. Czarnecki: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Diff. Equations 179 (2002) 278–310.https://doi.org/10.1006/jdeq.2001.4034Google Scholar
  10. F. Battahi, Z. Chbani, H. Riahi: Fast rates and strong convergence towards the minimum norm solution of dynamic trajectories with Tikhonov regularization for convex minimization under linear constraints, Evol. Equ. Control Theory 17 (2026) 259–281.https://doi.org/10.3934/eect.2025085Google Scholar
  11. F. Battahi, Z. Chbani, H. Riahi: On the simultaneous convergence of values and trajectories of continuous inertial dynamics with Tikhonov regularization to solve convex minimization with affine constraints, Appl. Set-Valued Analysis Optim. 7 (2025) 373–392.https://doi.org/10.23952/asvao.7.2025.3.06Google Scholar
  12. H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York (2017).https://doi.org/10.1007/978-3-319-48311-5Google Scholar
  13. J. Bolte, T. P. Nguyen, J. Peypouquet, B. W. Suter: From error bounds to the complexity of first-order descent methods for convex functions, Math. Programming 165 (2017) 471–507.https://doi.org/10.1007/s10107-016-1091-6Google Scholar
  14. R. I. Boţ: Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems Vol. 637, Springer, New York (2010).https://doi.org/10.1007/978-3-642-04900-2Google Scholar
  15. R. I. Boţ, D.-K. Nguyen: Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals, arXiv: 2406.00852 (2024).Google Scholar
  16. H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies, North-Holland, Amsterdam (1973).Google Scholar
  17. F. E. Browder: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Symp. Pure Math. Vol. 18, Part 2, American Mathematical Society, Providence (1976).https://doi.org/10.1090/pspum/018.2Google Scholar
  18. R. E. Bruck: A strongly convergent iterative solution of 0 ∈ U (x) for a maximal monotone operator U in Hilbert space, J. Math. Analysis Appl. 48 (1974) 114–126.https://doi.org/10.1016/0022-247X(74)90219-4Google Scholar
  19. R. Cominetti, J. Peypouquet, S. Sorin: Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization, J. Diff. Equations 245 (2008) 3753–3763.https://doi.org/10.1016/j.jde.2008.08.007Google Scholar
  20. M. G. Crandall, A. Pazy: Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57–94.https://doi.org/10.1007/BF02761448Google Scholar
  21. I. Ekeland, R. Témam: Convex Analysis and Variational Problems, Classics in Applied Mathematics, SIAM, Philadelphia (1999).https://doi.org/10.1137/1.9781611971088Google Scholar
  22. H. Furuya, K. Miyashiba, N. Kenmochi: Asymptotic behavior of solutions to a class of nonlinear evolution equations, J. Diff. Equations 62 (1986) 73–94.https://doi.org/10.1016/0022-0396(86)90106-3Google Scholar
  23. A. Haraux: Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, Masson, Paris (1991).Google Scholar
  24. M. M. Israel Jr., S. Reich: Asymptotic behavior of solutions of a nonlinear evolution equation, J. Math. Analysis Appl. 83 (1981) 43–53.https://doi.org/10.1016/0022-247X(81)90245-6Google Scholar
  25. N. Kenmochi: Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Ed. Chiba Univ. 30 (1981) 1–87.Google Scholar
  26. T. Kose: Solutions of saddle value problems by differential equations, Econometrica 24 (1956) 59–70.https://doi.org/10.2307/1905259Google Scholar
  27. S. K. Niederländer: Second-order dynamics with Hessian-driven damping for linearly constrained convex minimization, SIAM J. Control Optim. 59 (2021) 3708–3736.https://doi.org/10.1137/20M1323679Google Scholar
  28. S. K. Niederländer: On the Arrow-Hurwicz differential system for linearly constrained convex minimization, Optimization 73 (2023) 2313–2345.https://doi.org/10.1080/02331934.2023.2215799Google Scholar
  29. S. Reich: Nonlinear evolution equations and nonlinear ergodic theorems, Nonlinear Analysis 1 (1977) 319–330.https://doi.org/10.1016/S0362-546X(97)90001-8Google Scholar
  30. R. T. Rockafellar: Monotone operators associated with saddle-functions and minimax problems, in: Nonlinear Functional Analysis, F. E. Browder (ed.), Proc. Symp. Pure Math. Vol. 18, Part 1, American Mathematical Society (1970) 241–250.https://doi.org/10.1090/pspum/018.1/0285942Google Scholar
  31. R. T. Rockafellar: Saddle-points and convex analysis, in: Differential Games and Related Topics, H. W. Kuhn et al. (eds.), North-Holland, Amsterdam (1971) 109–127.Google Scholar
  32. A. N. Tikhonov, V. Y. Arsénine: Méthodes de Résolution de Problèmes Mal Posés, Mir, Moscow (1974).Google Scholar
  33. D. Torralba: Convergence Épigraphique et Changements d’Échelle en Analyse Variationnelle et Optimisation, Ph.D. thesis, Université de Montpellier II (1996).Google Scholar

Suggested citation

F. Battahi, Z. Chbani, S. K. Niederländer, H. Riahi. “Asymptotic Behavior of the Arrow-Hurwicz Differential System with Tikhonov Regularization.” Journal of Convex Analysis 33 (2026), No. 3&4, 565–596. https://doi.org/10.68381/jca33036

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