Journal of Global Optimization

, Volume 36, Issue 2, pp 191–217 | Cite as

Linearly Constrained Global Optimization and Stochastic Differential Equations

  • Panos Parpas
  • Berç Rustem
  • Efstratios N. Pistikopoulos
Original Paper

Abstract

A stochastic algorithm is proposed for the global optimization of nonconvex functions subject to linear constraints. Our method follows the trajectory of an appropriately defined Stochastic Differential Equation (SDE). The feasible set is assumed to be comprised of linear equality constraints, and possibly box constraints. Feasibility of the trajectory is achieved by projecting its dynamics onto the set defined by the linear equality constraints. A barrier term is used for the purpose of forcing the trajectory to stay within the box constraints. Using Laplace’s method we 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 projected diffusion process and show that its weak limit is given by Π. Numerical experiments using standard test problems from the literature are reported. Our results suggest that the method is robust and applicable to large-scale problems.

Keywords

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

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References

  1. 1.
    Aluffi-Pentini F., Parisi V., Zirilli F. (1985). Global optimization and stochastic differential equations. J. Optim. Theory Appl. 47(1):1–16CrossRefGoogle 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), 1988Google Scholar
  3. 3.
    Bender C.M., Orszag S.A. (1999). Advanced mathematical methods for scientists and engineers: asymptotic methods and perturbation theory. Springer-Verlag, BerlinGoogle Scholar
  4. 4.
    Chiang T.S., Hwang C.R., Sheu S.J. (1987). Diffusion for global optimization in R n. SIAM J. Control Optim. 25(3):737–753CrossRefGoogle Scholar
  5. 5.
    Floudas C.A., Pardalos P.M. (1990). A collection of test problems for constrained global optimization algorithms. Lecture Notes in Computer Science. vol. 455, Springer-Verlag, BerlinGoogle Scholar
  6. 6.
    Garabedian P.R. (1964). Partial differential equations. Wiley, New YorkGoogle Scholar
  7. 7.
    Gard T.C. (1988). Introduction to stochastic differential equations. Marcel Dekker Inc., New YorkGoogle Scholar
  8. 8.
    Gelfand S.B., Mitter S.K. (1991). Recursive stochastic algorithms for global optimization in R d. SIAM J. Control Optim. 29(5):999–1018CrossRefGoogle Scholar
  9. 9.
    Geman S., Hwang C.R. (1986). Diffusions for global optimization. SIAM J. Control Optim. 24(5):1031–1043CrossRefGoogle Scholar
  10. 10.
    Gidas, B.: The Langevin equation as a global minimization algorithm. In Disordered systems and biological organization (Les Houches, 1985), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 20. pp. 321–326. Springer, Berlin (1986)Google Scholar
  11. 11.
    Gidas, B.: Simulations and global optimization. In Random media (Minneapolis, Minn., 1985). IMA Vol. Math. Appl., vol. 7. pp. 129–145. Springer, New York (1987)Google Scholar
  12. 12.
    Gidas, B.: Metropolis-type Monte Carlo simulation algorithms and simulated annealing. In: Topics in contemporary probability and its applications, Probab. Stochastics Ser., pp. 159–232. CRC, Boca Raton, FL (1995).Google Scholar
  13. 13.
    Hwang C.R. (1980). Laplace’s method revisited: weak convergence of probability measures. Ann. Probab. 8(6):1177–1182CrossRefGoogle Scholar
  14. 14.
    Kushner H.J. (1987). Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via Monte Carlo. SIAM J. Appl. Math. 47(1):169–185CrossRefGoogle Scholar
  15. 15.
    Li-Zhi L., Liqun Q., Hon W.T. (2005). A gradient-based continuous method for large-scale optimization problems. J. Global Optim. 31(2):271CrossRefGoogle Scholar
  16. 16.
    Luenberger D.G. (1972). The gradient projection method along geodesics. Manage Sci. 18:620–631CrossRefGoogle Scholar
  17. 17.
    Luenberger D.G. (2003). Linear and nonlinear programming. Kluwer Academic Publishers, Boston, MA, 2nd edn.Google Scholar
  18. 18.
    Martin O.C., Monasson R., Zecchina R. (2001). Statistical mechanics methods and phase transitions in optimization problems. Theoret. Comput. Sci. 265(1–2):3–67CrossRefGoogle Scholar
  19. 19.
    Parpas, P., Rustem, B.: Global optimization of the scenario generation and portfolio selection problems. International Conference on Computational Science and Its Applications (ICCSA) 2006, to appear in Lecture Notes in Computer Science.Google Scholar
  20. 20.
    Pintér, J.: Global optimization: software, test problems, and applications. In: Handbook of Global Optimization, Vol. 2. Nonconvex Optim. Appl., vol. 62. pp. 515–569. Kluwer Acad. Publ. Dordrecht (2002)Google Scholar
  21. 21.
    Recchioni M.C., Scoccia A. (2000). A stochastic algorithm for constrained global optimization. J. Global Optim. 16(3):257–270CrossRefGoogle Scholar
  22. 22.
    Slotine J.J., Weiping L. (1991). Applied nonlinear control. Prentice-Hall International, Inc., London, UKGoogle Scholar
  23. 23.
    Zirilli, F.: The use of ordinary differential equations in the solution of nonlinear systems of equations. In: Nonlinear Optimization, 1981 (Cambridge, 1981), NATO Conf. Ser. II: Systems Sci., pp. 39–46. Academic Press, London (1982)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Panos Parpas
    • 1
  • Berç Rustem
    • 1
  • Efstratios N. Pistikopoulos
    • 2
  1. 1.Department of ComputingImperial CollegeLondonUK
  2. 2.Department of Chemical Engineering, Center for Process Systems EngineeringImperial CollegeLondonUK

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