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Article Dans Une Revue Journal of Optimization Theory and Applications Année : 2014

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

Résumé

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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Dates et versions

hal-02072808 , version 1 (19-03-2019)

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Boushra Abbas, Benar Fux Svaiter, Hedy Attouch. Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces. Journal of Optimization Theory and Applications, 2014, 161 (2), pp.331-360. ⟨10.1007/s10957-013-0414-5⟩. ⟨hal-02072808⟩
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