Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 161, Issue 2, Pages 331-360Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-013-0414-5
Keywords
Monotone inclusions; Newton method; Levenberg-Marquardt regularization; Dissipative dynamical systems; Lyapunov analysis; Weak asymptotic convergence; Forward-backward algorithms; Gradient-projection methods
Funding
- CNPq [302962/2011-5, 474944/2010-7, 480101/2008-6, 303583/2008-8]
- FAPERJ [E-26/102.940/2011, E-26/102.821/2008]
- PRONEX-Optimization
- [ANR-08-BLAN-0294-03]
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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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