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A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

H. Attouch (attouch***at***math.univ-montp2.fr)
B. F. Svaiter (benar***at***impa.br)

Abstract: We introduce non-autonomous continuous dynamical systems which are linked to Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant to Cauchy-Lipschitz theorem. By using Lyapunov analysis, we prove that their trajectories asymptotically weakly converge to equilibria. Discrete time version of these results provides new insight on Newton's method for solving monotone inclusions.

Keywords: Maximal monotone operators, Newton-like algorithm, Levenberg-Marquardt algorithm, non-autonomous differential equations, absolutely continuous trajectories, dissipative dynamical systems, Lyapunov analysis, weak asymptotic convergence, numerical convex optimization

Category 1: Complementarity and Variational Inequalities

Category 2: Infinite Dimensional Optimization

Category 3: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Citation: Submitted in Jan 27, 2010 to SIAM Journal on Control and Optimization

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Entry Submitted: 12/15/2010
Entry Accepted: 12/15/2010
Entry Last Modified: 12/15/2010

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