Set-Valued Analysis

, Volume 9, Issue 1–2, pp 3–11 | Cite as

An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping

  • Felipe Alvarez
  • Hedy Attouch


The ‘heavy ball with friction’ dynamical system x + γx + ∇f(x)=0 is a nonlinear oscillator with damping (γ>0). It has been recently proved that when H is a real Hilbert space and f: HR is a differentiable convex function whose minimal value is achieved, then each solution trajectory tx(t) of this system weakly converges towards a solution of ∇f(x)=0. We prove a similar result in the discrete setting for a general maximal monotone operator A by considering the following iterative method: xk+1x k −α k (x k xk−1)+λ k A(xk+1)∋0, giving conditions on the parameters λ k and α k in order to ensure weak convergence toward a solution of 0∈A(x) and extending classical convergence results concerning the standard proximal method.

Hilbert space monotone operator nonlinear oscillator with damping proximal iteration weak convergence Opial's lemma 


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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Felipe Alvarez
    • 1
  • Hedy Attouch
    • 2
  1. 1.Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMR 2071)Universidad de ChileSantiago
  2. 2.Laboratoire ACSIOM (CNRS EP 2066), Département de MathématiquesUniversité de Montpellier IIMontpellier Cedex 05France

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