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

B. Abbas (), H. Attouch () and Benar F. Svaiter ()
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B. Abbas: Université Montpellier II
H. Attouch: Université Montpellier II
Benar F. Svaiter: IMPA

Journal of Optimization Theory and Applications, 2014, vol. 161, issue 2, No 1, 360 pages

Abstract: 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.

Keywords: Monotone inclusions; Newton method; Levenberg–Marquardt regularization; Dissipative dynamical systems; Lyapunov analysis; Weak asymptotic convergence; Forward-backward algorithms; Gradient-projection methods (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (9)

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DOI: 10.1007/s10957-013-0414-5

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