Advertisement

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
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antipin, A. S.: Minimization of convex functions on convex sets by means of differential equations, Differential Equations 30(9) (1994), 1365–1375.Google Scholar
  2. 2.
    Alvarez, F.: On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim. 38(4) (2000), 1102–1119.Google Scholar
  3. 3.
    Attouch, H. and Alvarez, F.: The heavy ball with friction dynamical system for convex constrained minimization problems, In: Lecture Notes in Econom. Math. Systems 481, Springer, Berlin, 2000, pp. 25–35.Google Scholar
  4. 4.
    Attouch, H., Goudou, X. and Redont, P.: The heavy ball with friction method. I. The continuous dynamical system, Comm. Contemp. Math. 2(1) (2000), 1–34.Google Scholar
  5. 5.
    Brezis, H.: Opérateurs maximaux monotones, Math. Stud. 5, North-Holland, Amsterdam, 1973.Google Scholar
  6. 6.
    Brezis, H.: Asymptotic Behaviour of Some Evolution Systems: Nonlinear Evolution Equations, Academic Press, New York, 1978.Google Scholar
  7. 7.
    Bruck, R. E.: Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal. 18 (1975), 15–26.Google Scholar
  8. 8.
    Güler, O.: On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29 (1991), 403–419.Google Scholar
  9. 9.
    Jules, F. and Mainge, P. E.: Numerical approach to stationary solution of a second order dissipative dynamical system, Preprint 99/01, Département de Mathématiques et Informatique, U. des Antilles et de la Guyane, 1999.Google Scholar
  10. 10.
    Martinet, B.: Régularisation d'inéquations variationnelles par approximations successives, Rev. Française d'Informatique et Recherche Operationnelle 4 (1970), 154–159.Google Scholar
  11. 11.
    Martinet, B.: Algorithmes pour la résolution de problèmes d'optimisation et de minimax, PhD Thesis, U. Grenoble, 1972.Google Scholar
  12. 12.
    Minty, G.: Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346.Google Scholar
  13. 13.
    Moreau, J. J.: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273–299.Google Scholar
  14. 14.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.Google Scholar
  15. 15.
    Polyak, B. T.: Some methods of speeding up the convergence of iterative methods, Zh. Vychisl. Mat. Mat. Fiz. 4 (1964), 1–17.Google Scholar
  16. 16.
    Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM J. Control Optim. (1976), 877–898.Google Scholar

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

Personalised recommendations