Journal of Global Optimization

, Volume 45, Issue 1, pp 95–110 | Cite as

Convergence analysis of a global optimization algorithm using stochastic differential equations

  • Panos Parpas
  • Berç Rustem


We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.


Stochastic global optimization Simulated annealing Stochastic differential equations Fokker-Planck equation Laplace’s method Projection algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aluffi-Pentini F., Parisi V., Zirilli F.: Global optimization and stochastic differential equations. J. Optim. Theory Appl. 47(1), 1–16 (1985)CrossRefGoogle Scholar
  2. 2.
    Aluffi-Pentini F., Parisi V., Zirilli F.: A global optimization algorithm using stochastic differential equations. ACM Trans. Math. Softw. 14(4), 345–365 (1989)CrossRefGoogle Scholar
  3. 3.
    Bender C.M., Orszag S.A.: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Springer-Verlag, Berlin (1999)Google Scholar
  4. 4.
    Chiang T.S., Hwang C.R., Sheu S.J.: Diffusion for global optimization in R n. SIAM J. Control Optim. 25(3), 737–753 (1987)CrossRefGoogle Scholar
  5. 5.
    Gelfand S.B., Mitter S.K.: Recursive stochastic algorithms for global optimization in R d. SIAM J. Control Optim. 29(5), 999–1018 (1991)CrossRefGoogle Scholar
  6. 6.
    Geman S., Hwang C.R.: Diffusions for global optimization. SIAM J. Control Optim. 24(5), 1031–1043 (1986)CrossRefGoogle Scholar
  7. 7.
    Gidas, B.: The Langevin equation as a global minimization algorithm. In: Disordered Systems and Biological Organization (Les Houches, 1985), Vol. 20 of NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., pp. 321–326. Springer, Berlin (1986)Google Scholar
  8. 8.
    Gidas, B.: Simulations and global optimization. In: Random Media (Minneapolis, Minn., 1985), Vol. 7 of IMA Vol. Math. Appl., pp. 129–145. Springer, New York (1987)Google Scholar
  9. 9.
    Gidas, B.: Metropolis-type Monte Carlo simulation algorithms and simulated annealing. In: Topics in Contemporary Probability and Its Applications, Probability Stochastics Series, pp. 159–232. CRC, Boca Raton, FL (1995)Google Scholar
  10. 10.
    Hwang C.R.: Laplace’s method revisited: weak convergence of probability measures. Ann. Probab. 8(6), 1177–1182 (1980)CrossRefGoogle Scholar
  11. 11.
    Kushner H.J.: Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via Monte Carlo. SIAM J. Appl. Math. 47(1), 169–185 (1987)CrossRefGoogle Scholar
  12. 12.
    Li-Zhi L., Liqun Q., Hon W.T.: A gradient-based continuous method for large-scale optimization problems. J. Glob. Optim. 31(2), 271 (2005)CrossRefGoogle Scholar
  13. 13.
    Luenberger D.G.: The gradient projection method along geodesics. Manag. Sci. 18, 620–631 (1972)CrossRefGoogle Scholar
  14. 14.
    Luenberger D.G.: Linear and Nonlinear Programming, 2nd edn. Kluwer Academic Publishers, Boston (2003)Google Scholar
  15. 15.
    Maringer, D., Parpas, P.: Global optimization of higher order moments in portfolio selection. J. Glob. Optim. doi: 10.1007/s10898-007-9224-3
  16. 16.
    Oksendal, B.: Stochastic Differential Equations, an Introduction with Applications, 6th edn. Springer, New YorkGoogle Scholar
  17. 17.
    Parpas, P., Rustem, B., Pistikopoulos, E.N.: Global optimization of robust chance constrained problems. J. Glob. Optim. doi: 10.1007/s10898-007-9244-z
  18. 18.
    Parpas P., Rustem B., Pistikopoulos E.N.: Linearly constrained global optimization and stochastic differential equations. J. Glob. Optim. 36(2), 191–217 (2006)CrossRefGoogle Scholar
  19. 19.
    Recchioni M.C., Scoccia A.: A stochastic algorithm for constrained global optimization. J. Glob. Optim. 16(3), 257–270 (2000)CrossRefGoogle Scholar
  20. 20.
    Zirilli, F.: The use of ordinary differential equations in the solution of nonlinear systems of equations. In: Nonlinear Optimization, 1981 (Cambridge, 1981), NATO Conference Series II: Systems Science, pp. 39–46. Academic Press, London (1982)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of ComputingImperial CollegeLondonUK

Personalised recommendations