Skip to main content
Log in

The gradient and heavy ball with friction dynamical systems: the quasiconvex case

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the gradient system \(\dot x(t)+\nabla\phi(x(t))=0\) and the so-called heavy ball with friction dynamical system \(\ddot x(t) +\lambda\dot x(t)+\nabla\phi(x(t))=0\) , as well as an implicit discrete (proximal) version of it, and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function Φ. Minimization properties of trajectories are obtained under various additional assumptions. We finally show a minimizing property of the heavy ball method which is not shared by the gradient method: the genericity of the convergence of each trajectory, at least when Φ is a Morse function, towards local minimum of Φ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alvarez F. (2000). On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38(4): 1102–1119

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvarez F. and Attouch H. (2001). An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping. Wellposedness in optimization and related topics. Set Valued Anal. 9(1–2): 3–11

    Article  MathSciNet  MATH  Google Scholar 

  3. Arrow K. and Debreu G. (1954). Existence of an equilibrium for a competitive economy. Econometrica 22: 265–290

    Article  MathSciNet  MATH  Google Scholar 

  4. Attouch H., Cabot A. and Redont P. (2002). The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations. Adv. Math. Sci. Appl. 12(1): 273–306

    MathSciNet  MATH  Google Scholar 

  5. Attouch H. and Czarnecki M. (2002). Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differ. Equ. 179: 278–310

    Article  MathSciNet  MATH  Google Scholar 

  6. Attouch H., Goudou X. and Redont P. (2000). The heavy ball with friction method, I. The continuous dynamical system: global exploration of the local minima of real-valued function by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 2(1): 1–34

    Article  MathSciNet  MATH  Google Scholar 

  7. Attouch, H., Soubeyran, A.: Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex Anal. 13(2), 207–224 (2006, to appear)

    Google Scholar 

  8. Attouch H. and Teboule M. (2004). Regularized Lotka–Volterra dynamical system as continuous proximal-like method in optimization. J. Optim. Theory Appl. 121(3): 77–106

    Article  MathSciNet  Google Scholar 

  9. Aussel D. (1998). Subdifferential properties of quasiconvex and pseudoconvex function, unified approach. J. Optim. Theory Appl. 97: 29–45

    Article  MathSciNet  MATH  Google Scholar 

  10. Aussel, D.: Contributions en analyse multivoque et en optimisation, thèse hdr. Ph.D. thesis, Université Montpellier 2 (2005)

  11. Aussel D. and Daniilidis A. (2000). Normal characterization of the main classes of quasiconvex analysis. Set Valued Anal. 8: 219–236

    Article  MathSciNet  MATH  Google Scholar 

  12. Baillon, J.: Un exemple concernant le comportement asymptotique de la solution du problème \(du/dt+\partial\varphi(u)\ni 0\)J. Funct. Anal. 28, 369–376 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Barron N. and Liu W. (1997). Calculus of variation in L . Appl. Math. Optim. 35: 237–263

    MathSciNet  MATH  Google Scholar 

  14. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contraction. Lectures Notes No. 5, North Holland (1973)

  15. Bruck R. (1975). Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18: 15–26

    Article  MathSciNet  MATH  Google Scholar 

  16. Brunovsky P. and Polàcik P. (1997). The morse-smale structure of generic reaction-diffusion equation in higher space dimension. J. Differ. Equ. 135(1): 129–181

    Article  MATH  Google Scholar 

  17. Crouzeix, J.: A review of continuity and differentiability properties of quasiconvex functions on \({\mathbb{R}}^n\) . Research Notes in Mathematics 57, Aubin and Vinter Ed, pp. 18–34 (1982)

  18. Crouzeix, J.P.: Contribution à l’étude des fonctions quasiconvexes. Ph.D. Thesis, Université de Clermont-Ferrand Thèse d’Etat (1977)

  19. Crouzeix, J.-P.: La convexité généralisée en économie mathématique. In: Penot, J. (ed.) Proceedings of 2003 MODE-SMAI Conference, vol. 13, pp. 31–40 (2003)

  20. Dacorogna B. (1989). Direct methods in the calculus of variation. Applied Mathematical Sciences, vol. 78. Springer, Heidelberg

    Google Scholar 

  21. Debreu G. (1959). Theory of Value. Wiley, New York

    MATH  Google Scholar 

  22. Debreu G. (1983). Mathematical Economics: Twenty papers. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  23. Dussault J.P. (2000). Convergence of implementable descent algorithms for unconstrained optimization. J. Optim. Theory Appl. 104(3): 739–745

    Article  MathSciNet  MATH  Google Scholar 

  24. Ginsberg W. (1973). Concavity and quasiconcavity in economics. J. Econ. Theory 6: 596–605

    Article  MathSciNet  Google Scholar 

  25. Goudou X. and Munier J. (2005). Asymptotic behavior of solutions of a gradient-like integrodifferential Volterra inclusion. Adv. Math. Sci. Appl. 15: 509–525

    MathSciNet  MATH  Google Scholar 

  26. Guler O. (1991). On the convergence of the proximal point algorithm for convex optimization. SIAM J. Control Appl. 29(2): 403–419

    Article  MathSciNet  Google Scholar 

  27. Haraux A. (1990). Systèmes dynamiques dissipatifs et applications. Masson, Paris

    Google Scholar 

  28. Haraux A. and Jendoubi M. (1998). Convergence of solutions of second-order gradient-like systems with analytic nonlinearities. J. Differ. Equ. 144(2): 313–320

    Article  MathSciNet  MATH  Google Scholar 

  29. Jendoubi, M., Polacik, P.: Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping. In: Proceedings Section A: Mathematics, Royal Society of Edinburgh, pp. 1137–1153(17) (2003). http://www.ingentaconnect.com/content/rse/proca/2003/00000133/00000005/art00007

  30. Kiwiel K. and Murty K. (1996). Convergence of the steepest descent method for minimizing quasiconvex functions. J. Optim. Theory Appl. 89(1): 221–226

    Article  MathSciNet  MATH  Google Scholar 

  31. Lemaire, B.: New methods in optimization and their industrial uses. In: Penot, J. (ed.) Inter. Series of Numerical Math., vol. 87. Birkhauser-Verlag, Basel, pp. 73–87. Symp. Pau and Paris (1989)

  32. Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française d’Inform. Recherche Opérationnelle, Série R-3 4, 154–158 (1970)

    Google Scholar 

  33. Martinez-Legaz J.E. (2005). Generalized convex duality and its economic applications, handbook of generalized convexity and generalized monotonicity. Nonconvex Optim. Appl. 76: 237–292

    Article  MathSciNet  Google Scholar 

  34. Opial Z. (1967). Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73: 591–597

    Article  MathSciNet  MATH  Google Scholar 

  35. Perko L. (1996). Differential equations and dynamical systems. Springer, Heidelberg

    MATH  Google Scholar 

  36. Polyack B. (1964). Some methods of speeding up the convergence of iterative methods. Z. Vylist Math. Fiz. 4: 1–17

    Google Scholar 

  37. Rockafellar R. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14: 877–898

    Article  MathSciNet  MATH  Google Scholar 

  38. Seeger A. and Volle M. (1995). On a convolution operation obtained by adding level sets: classical and new results. Oper. Res. 29(2): 131–154

    MathSciNet  MATH  Google Scholar 

  39. Takayama A. (1995). Mathematical Economics. Cambridge University Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to X. Goudou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goudou, X., Munier, J. The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math. Program. 116, 173–191 (2009). https://doi.org/10.1007/s10107-007-0109-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-007-0109-5

Keywords

Mathematics Subject Classification (2000)

Navigation