The Second-order in Time Continuous Newton Method

  • H. Attouch
  • P. Redont


Let H be a real Hilbert space and Ф : H ↦ ℝ a twice continuously differentiable function, whose Hessian is Lipschitz continuous on bounded sets. We study the Newton-like second-order in time nonlinear dissipative dynamical system:\(\ddot x\left( t \right) + {\nabla ^2}\Phi \left( {x\left( t \right)} \right)\dot x\left( t \right) + \nabla \Phi \left( {x\left( t \right)} \right) = 0 \), plus Cauchy data, mainly in view of the unconstrained minimization of the function Ф.The main result is the gradient vanishing along any bounded trajectory as time goes to infinity. Results concerning the convergence of every bounded solution to a critical point are given in peculiar situations: when Ф is convex (with only one minimum) or is a Morse function.


dissipative dynamical system, optimization local minima, convex minimization, asymptotic behaviour, Newton method 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • H. Attouch
    • 1
  • P. Redont
    • 1
  1. 1.ACSIOM-CNRS EP 2066, Département de Mathématiques, case 51Université Montpellier IIMontpellier cedex 5France

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