A second-order gradient-like dissipative dynamical system with Hessian-driven damping.

Álvarez, Felipe; Attouch, Hédy; Bolte, Jérôme; Redont, Patrick

doi:10.1016/s0021-7824(01)01253-3

Cited by 155 publications

(210 citation statements)

References 17 publications

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“…In the case of convex optimization, it bears interesting connections with the second-order continuous dynamic approach developed by Alvarez, Attouch, Bolte, and Redont in [3], see also [4], [6] (Newton's dynamic is regularized by adding an inertial term, and a viscous damping term, which provides a second-order dissipative dynamical system with Hessian-driven damping.) Another interesting regularization method (based on the regularization of the objective function) has been developed by Alvarez and Pérez in [2].…”

Section: Introductionmentioning

confidence: 99%

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

Attouch¹,

Svaiter²

2011

SIAM J. Control Optim.

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132

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We introduce non-autonomous continuous dynamical systems which are linked to the Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on the 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 the Cauchy-Lipschitz theorem. By using Lyapunov methods, we prove that their trajectories converge weakly to equilibria. Time discretization of these dynamics gives algorithms providing new insight into Newton's method for solving monotone inclusions.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

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Section: Introductionmentioning

confidence: 99%

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

Attouch¹,

Svaiter²

2011

SIAM J. Control Optim.

Self Cite

132

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“…We let Q = Id. Assume that all the eigenvalues of P are real and larger than 1 2 . Then, the vector u defined the differential system (2.17) converges to P −1 b as t → +∞.…”

Section: Convergence Resultsmentioning

confidence: 99%

Differential equations and solution of linear systems

Chehab

Laminie

2005

Numer Algor

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Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretization of differential systems. In this paper, we propose to derive iterative schemes for solving linear systems of equations by modeling the problem to solve as a stable state of a proper differential system; the solution of the original linear problem is then computed numerically by applying a time marching scheme. We discuss some aspects of this approach, which allows to recover some known methods but also to introduce new ones. We give convergence results and numerical illustrations.

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“…The first convergence result for gradient systems is due to Łojasiewicz himself [15]; he also proved that every real analytic function satisfies the Łojasiewicz gradient inequality (4). Later, Haraux and Jendoubi [9] and Alvarez et al [2] proved convergence results for damped second order ordinary differential equations by using a gradient inequality for a natural Lyapunov function. More recently, it has been shown that many convergence results can be unified into a general theorem for the ordinary differential equation (1), provided that that equation admits a strict Lyapunov function, the Lyapunov function satisfies a gradient inequality and an angle condition holds between E and F [1,7,12].…”

Section: Introductionmentioning

confidence: 99%