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.
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Acknowledgements
H. Attouch is partially supported by ANR-08-BLAN-0294-03; Benar F. Svaiter is partially supported by CNPq grants 302962/2011-5, 474944/2010-7, 480101/2008-6, 303583/2008-8, FAPERJ grants E-26/102.940/2011, E-26/102.821/2008, and by PRONEX-Optimization.
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Abbas, B., Attouch, H. & Svaiter, B.F. Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces. J Optim Theory Appl 161, 331–360 (2014). https://doi.org/10.1007/s10957-013-0414-5
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DOI: https://doi.org/10.1007/s10957-013-0414-5