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Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics

Author

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  • Hedy Attouch

    (Université Montpellier)

  • Zaki Chbani

    (Cadi Ayyad University)

  • Jalal Fadili

    (ENSICAEN, UNICAEN, CNRS, GREYC)

  • Hassan Riahi

    (Cadi Ayyad University)

Abstract

In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are, respectively, associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01859-2
    DOI: 10.1007/s10957-021-01859-2
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    3. Myeongmin Kang & Myungjoo Kang & Miyoun Jung, 2015. "Inexact accelerated augmented Lagrangian methods," Computational Optimization and Applications, Springer, vol. 62(2), pages 373-404, November.
    4. Damek Davis & Wotao Yin, 2017. "Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 783-805, August.
    5. Boţ, Radu Ioan & Csetnek, Ernö Robert & Hendrich, Christopher, 2015. "Inertial Douglas–Rachford splitting for monotone inclusion problems," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 472-487.
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    Cited by:

    1. Samir Adly & Hedy Attouch & Van Nam Vo, 2023. "Convergence of Inertial Dynamics Driven by Sums of Potential and Nonpotential Operators with Implicit Newton-Like Damping," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 290-331, July.
    2. Ke-wei Ding & Jörg Fliege & Phan Tu Vuong, 2025. "Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem," Computational Optimization and Applications, Springer, vol. 90(1), pages 151-192, January.

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