Advertisement

A multi-sequential number-theoretic optimization algorithm using clustering methods

  • Xu Qing-song Email author
  • Liang Yi-zeng 
  • Hou Zhen-ting 
Mathematics
  • 41 Downloads

Abstract

A multi-sequential number-theoretic optimization method based on clustering was developed and applied to the optimization of functions with many local extrema. Details of the procedure to generate the clusters and the sequential schedules were given. The algorithm was assessed by comparing its performance with generalized simulated annealing algorithm in a difficult instructive example and a D-optimum experimental design problem. It is shown the presented algorithm to be more effective and reliable based on the two examples.

Key words

retention ratio cluster contraction local search 

CLC number

O29 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Avriel M. Nonlinear Programming Analysis and Methods [M]. Englewood Cliffs: Prentice-Hall, 1976.zbMATHGoogle Scholar
  2. [2]
    Nash J C. Nonlinear Parameter Estimation: an Intergrated System in BASIC [M]. New York: Deker, 1987.Google Scholar
  3. [3]
    Horst R, Tuy H. Global Optimization [M]. Berlin: Springer, 1990.CrossRefGoogle Scholar
  4. [4]
    Bertmas D, Tsitsiklis J. Simulated annealing [J]. Statis Sci, 1993, 8(1): 10–15.CrossRefGoogle Scholar
  5. [5]
    Rubinstain R Y. Monte Carlo Optimization, Simulation and Sensitivity of Queuing Networks [M]. New York: Wiley, 1986.Google Scholar
  6. [6]
    Niederreiter H. A quasi-monte carlo method for the approximate of extreme values of a function[A]. Studies in Pure Math[C]. Basel: Birkhauser, 1983.zbMATHGoogle Scholar
  7. [7]
    Fang K T, Wang Y. A unified approach to maximum likelihood estimation [J]. Chinese J Appl Probab Statist, 1990, 6(4): 412–418. (in Chinese)MathSciNetGoogle Scholar
  8. [8]
    Niederreider H, Peart P. Localization of search in quasi-monte carlo methods for global optimization [J]. SIAM J Sci Statist Comput, 1986, 7(6): 660–664.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Fang K T, Wang Y. A sequential algorithm for solving a system of nonlinear equations [J]. J Comput Math, 1991, 9(11): 9–16.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Zhang L, Liang Y Z, Yu R Q, at al. Sequential number-theoretic optimization method applied to chemical quantitative analysis [J]. J Chemometrics, 1997, 11(3): 267–281.CrossRefGoogle Scholar
  11. [11]
    Gong F, Cui H, Zhang L, et al. An improved algorithm of sequential number-theoretic optimization based on clustering technique [J]. Chemometrics Intel Lab Syst, 1999, 45(1–2): 339–346.CrossRefGoogle Scholar
  12. [12]
    Rinny Kan A H J, Timmer G T. Stochastic global optimization methods part II: multi lever methods [J]. Mathematical Programming, 1987, 39(1): 57–78.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Bohachevsky I O, Jonson M E, Stein M L. Generalized simulated annealing for function optimization [J]. Technometrics, 1986, 28(2): 209–217.CrossRefGoogle Scholar
  14. [14]
    Fang K T, Wang Y. Number-theoretic Methods in Statistics [M]. London: Chapman and Hall, 1993.Google Scholar
  15. [15]
    Halton J H. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimentional integrals [J]. Numer Math, 1960, 2(1): 84–90.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Nieterreider H. Quasi-monte carlo methods and pseudo-random numbers [J]. Bull Amer Math Soc, 1978, 84(4): 957–1041.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Rinny Kan A H J, Timmer G T. Stochastic global optimization methods part I: clustering methods [J]. Mathematical Programming, 1987, 39(1): 27–56.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Atkinson A C, Bogacka B. Bogacki, MB D-and T-optimum designs for the kinetics of a reversible chemical reaction [J]. Chemom Lab Syst, 1998, 43(2): 185–198.CrossRefGoogle Scholar
  19. [19]
    Xu Q S, Liang Y Z, Fang K T. The effects of different experimental designs on parameter estimation in the kinetics of a reversible chemical reaction [J]. Chemometrics Intel Lab Syst, 2000, 52(2): 155–166.CrossRefGoogle Scholar
  20. [20]
    Chernoff H. Locally optimal designs for estimating parameters [J]. Ann Math Statist, 1953, 24(4): 586–602.MathSciNetCrossRefGoogle Scholar

Copyright information

© Central South University 2005

Authors and Affiliations

  • Xu Qing-song 
    • 1
    Email author
  • Liang Yi-zeng 
    • 2
  • Hou Zhen-ting 
    • 1
  1. 1.School of Mathematical Science and Computing TechnologyCentral South UniversityChangshaChina
  2. 2.School of Chemistry and Chemical EngineeringCentral South UniversityChangshaChina

Personalised recommendations