The heavy ball with friction method, I. The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system

Communications in Contemporary Mathematics

Volumn 2, Issue 1, 2000, Pages 1-34

  Attouch, H ^a   Goudou, X ^a   Redont, P ^a  

^a CNRS   (France)

Author keywords

Asymptotic behaviour; Convex minimization; Dissipative dynamical system; Gradient system; Heavy ball with friction; Local minima; Morse function; Optimization

Indexed keywords

EID: 23044521290     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

1. R. Abraham and J. Robbin, Transversal Mappings and Flows, W. A. Benjamin, New York (1967).
2. F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space, preprint 98-05, Département de Mathématiques, Université Montpellier II, to appear in SIAM J. of Control and Optimization.
3. A. S. Antipin, Second order proximal differential systems with feedback control, Differential Equations 29 (1993) 1597-1607.
4. J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley (1984).
5. H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method, J. Diff. Eq. 128 (1996) 519-540.
6. H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method: II the discrete dynamical system, working paper, Département de Mathématiques, Université de Montpellier II.
7. J.-B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème du/dt + ∂φ(u) = 0, Journal of Functional Analysis 28 (1978) 369-376.
8. J. M. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Diff. Eq. 27 (1978) 224-265.
9. Bertsekas, Nonlinear Programming, MIT, Athena Scientific, Belmont, Massachussets (1995).
10. H. Brézis, Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations, in Contributions to Nonlinear Analysis, ed. E. H. Zarantonello, Academic Press, New York (1971) 101-156.
11. H. Brézis, Asymptotic behaviour of some evolution systems, Nonlinear evolution equations, Academic Press (1978).
12. R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, Journal of Functional Analysis 18 (1975) 15-26.
13. M. Chaperon and F. Coudray, Invariant manifolds conjugacies and blowup, Ergodic Theory and Dynamical Systems 17 (1997) 783-791.
14. C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations in Non-linear Evolution Equations, ed. M. G. Crandall, Academic Press, New York (1978), pp. 103-123.
15. H. Furuya, K. Miyashiba and N. Kenmochi, Asymptotic behaviour of solutions to a class of nonlinear evolution equations, J. Diff. Eq. 62 (1986) 73-94.
16. X. Goudou, Genericity of the convergence towards a local minimum of the heavy ball method, to appear.
17. J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Mathematical Surveys and Monographs, Vol. 25, A.M.S., Providence, RI (1987).
18. J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Diff. Eq. 73 (1988) 197-214.
19. A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, RMA 17, Masson, Paris (1991).
20. A. Haraux, Nonlinear Evolution Equations: Global Behaviour of Solutions, Lecture Notes in Math., Vol. 481, Springer (1981).
21. J. L. Lions, Perturbations Singulières Dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Math., Vol. 323, Springer (1973).
22. Z. Opial, Weak convergence, of the sequence of successive approximations for non-expansive mappings, Bull. of the Amer. Math. Soc. 73 (1967) 591-597.
23. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer, (1982).
24. L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Vol. 7, Springer, (1996).
25. B. T. Polyack, Some methods of speeding up the convergence of iterative methods, Z. Vylist Math. Fiz. 4 (1964) 1-17.
26. P. Redont, Equation de la boule pesante avec frottement: exemple de solution non convergente, Prépublication 99, Département de Mathématiques, Université de Montpellier II, http://www.math.univ-montp2.fr.
27. S. Smale, A convergent process of price adjustment and global Newton method, Journal of Mathematical Economics 3 (1976) 107-120.