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Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

Author

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  • B. Abbas

    (Université Montpellier II)

  • H. Attouch

    (Université Montpellier II)

  • Benar F. Svaiter

    (IMPA)

Abstract

In a Hilbert space setting we introduce dynamical systems, which are linked to Newton and Levenberg–Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M=A+B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy–Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions.

Suggested Citation

  • B. Abbas & H. Attouch & Benar F. Svaiter, 2014. "Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 331-360, May.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:2:d:10.1007_s10957-013-0414-5
    DOI: 10.1007/s10957-013-0414-5
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    References listed on IDEAS

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    1. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    2. H. Attouch & P. Redont & B. F. Svaiter, 2013. "Global Convergence of a Closed-Loop Regularized Newton Method for Solving Monotone Inclusions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 624-650, June.
    3. J. Bolte, 2003. "Continuous Gradient Projection Method in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 235-259, November.
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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. Boţ, Radu Ioan & Kanzler, Laura, 2021. "A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    3. Boţ, R.I. & Csetnek, E.R. & Vuong, P.T., 2020. "The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces," European Journal of Operational Research, Elsevier, vol. 287(1), pages 49-60.
    4. Samir Adly & Hedy Attouch & Manh Hung Le, 2024. "A Doubly Nonlinear Evolution System with Threshold Effects Associated with Dry Friction," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1188-1218, November.
    5. Lien T. Nguyen & Andrew Eberhard & Xinghuo Yu & Alexander Y. Kruger & Chaojie Li, 2024. "Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 844-879, October.
    6. Pankaj Gautam & Daya Ram Sahu & Avinash Dixit & Tanmoy Som, 2021. "Forward–Backward–Half Forward Dynamical Systems for Monotone Inclusion Problems with Application to v-GNE," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 491-523, August.
    7. Rieger, Janosch & Tam, Matthew K., 2020. "Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions," Applied Mathematics and Computation, Elsevier, vol. 381(C).

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