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Tikhonov Regularization of Second-order plus First-order Primal-dual Dynamical Systems for Separable Convex Optimization

Author

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  • Xiangkai Sun

    (Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, College of Mathematics and Statistics, Chongqing Technology and Business University)

  • Lijuan Zheng

    (Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, College of Mathematics and Statistics, Chongqing Technology and Business University)

  • Kok Lay Teo

    (School of Mathematical Sciences, Sunway University)

Abstract

This paper deals with a Tikhonov regularized second-order plus first-order primal-dual dynamical system with time scaling for separable convex optimization problems with linear equality constraints. This system consists of two second-order ordinary differential equations for the primal variables and one first-order ordinary differential equation for the dual variable. By utilizing the Lyapunov analysis approach, we obtain the convergence properties of the primal-dual gap, the objective function error, the feasibility measure and the gradient norm of the objective function along the trajectory. We also establish the strong convergence of the primal trajectory generated by the dynamical system towards the minimal norm solution of the separable convex optimization problem. Furthermore, we give numerical experiments to illustrate the theoretical results, showing that our dynamical system performs better than those in the literature in terms of convergence rates.

Suggested Citation

  • Xiangkai Sun & Lijuan Zheng & Kok Lay Teo, 2025. "Tikhonov Regularization of Second-order plus First-order Primal-dual Dynamical Systems for Separable Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 207(1), pages 1-36, October.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:1:d:10.1007_s10957-025-02766-6
    DOI: 10.1007/s10957-025-02766-6
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    References listed on IDEAS

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    1. Hedy Attouch & Zaki Chbani & Jalal Fadili & Hassan Riahi, 2022. "Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 704-736, June.
    2. Mikhail A. Karapetyants, 2024. "A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique," Computational Optimization and Applications, Springer, vol. 87(2), pages 531-569, March.
    3. Ernö Robert Csetnek & Mikhail A. Karapetyants, 2024. "Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1385-1420, September.
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