Advertisement

Journal of Optimization Theory and Applications

, Volume 79, Issue 1, pp 157–181 | Cite as

Lipschitzian optimization without the Lipschitz constant

  • D. R. Jones
  • C. D. Perttunen
  • B. E. Stuckman
Contributed Papers

Abstract

We present a new algorithm for finding the global minimum of a multivariate function subject to simple bounds. The algorithm is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. This is done by carrying out simultaneous searches using all possible constants from zero to infinity. On nine standard test functions, the new algorithm converges in fewer function evaluations than most competing methods.

The motivation for the new algorithm stems from a different way of looking at the Lipschitz constant. In particular, the Lipschitz constant is viewed as a weighting parameter that indicates how much emphasis to place on global versus local search. In standard Lipschitzian methods, this constant is usually large because it must equal or exceed the maximum rate of change of the objective function. As a result, these methods place a high emphasis on global search and exhibit slow convergence. In contrast, the new algorithm carries out simultaneous searches using all possible constants, and therefore operates at both the global and local level. Once the global part of the algorithm finds the basin of convergence of the optimum, the local part of the algorithm quickly and automatically exploits it. This accounts for the fast convergence of the new algorithm on the test functions.

Key Words

Global optimization Lipschitzian optimization space covering space partitioning 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Stuckman, B., Care, M., andStuckman, P.,System Optimization Using Experimental Evaluation of Design Performance, Engineering Optimization, Vol. 16, pp. 275–289, 1990.Google Scholar
  2. 2.
    Shubert, B.,A Sequential Method Seeking the Global Maximum of a Function, SIAM Journal on Numerical Analysis, Vol. 9, pp. 379–388, 1972.Google Scholar
  3. 3.
    Galperin, E.,The Cubic Algorithm, Journal of Mathematical Analysis and Applications, Vol. 112, pp. 635–640, 1985.Google Scholar
  4. 4.
    Pinter, J.,Globally Convergent Methods for n-Dimensional Multiextremal Optimization, Optimization, Vol. 17, pp. 187–202, 1986.Google Scholar
  5. 5.
    Horst, R., andTuy, H.,On the Convergence of Global Methods in Multiextremal Optimization, Journal of Optimization Theory and Applications, Vol. 54, pp. 253–271, 1987.Google Scholar
  6. 6.
    Mladineo, R.,An Algorithm for Finding the Global Maximum of a Multimodal, Multivariate Function, Mathematical Programming, Vol. 34, pp. 188–200, 1986.Google Scholar
  7. 7.
    Preparata, F., andShamos, M.,Computational Geometry: An Introduction, Springer-Verlag, New York, New York, 1985.Google Scholar
  8. 8.
    Dixon, L., andSzego, G.,The Global Optimization Problem: An Introduction, Toward Global Optimization 2, Edited by L. Dixon and G. Szego, North-Holland, New York, New York, pp. 1–15, 1978.Google Scholar
  9. 9.
    Yao, Y.,Dynamic Tunneling Algorithm for Global Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 19, pp. 1222–1230, 1989.Google Scholar
  10. 10.
    Stuckman, B., andEason, E.,A Comparison of Bayesian Sampling Global Optimization Techniques, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 22, pp. 1024–1032, 1992.Google Scholar
  11. 11.
    Belisle, C., Romeijn, H., andSmith, R.,Hide-and-Seek: A Simulated Annealing Algorithm for Global Optimization, Technical Report 90-25, Department of Industrial and Operations Engineering, University of Michigan, 1990.Google Scholar
  12. 12.
    Boender, C., et al.,A Stochastic Method for Global Optimization, Mathematical Programming, Vol. 22, pp. 125–140, 1982.Google Scholar
  13. 13.
    Snyman, J., andFatti, L.,A Multistart Global Minimization Algorithm with Dynamic Search Trajectories, Journal of Optimization Theory and Applications, Vol. 54, pp. 121–141, 1987.Google Scholar
  14. 14.
    Kostrowicki, J., andPiela, L.,Diffusion Equation Method of Global Minimization: Performance on Standard Test Functions, Journal of Optimization Theory and Applications, Vol. 69, pp. 269–284, 1991.Google Scholar
  15. 15.
    Perttunen, C.,Global Optimization Using Nonparametric Statistics, University of Louisville, PhD Thesis, 1990.Google Scholar
  16. 16.
    Perttunen, C., andStuckman, B.,The Rank Transformation Applied to a Multiunivariate Method of Global Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 20, pp. 1216–1220, 1990.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • D. R. Jones
    • 1
  • C. D. Perttunen
    • 2
  • B. E. Stuckman
    • 2
  1. 1.General Motors Research and Development CenterWarren
  2. 2.Brooks and Kushman, Department of Patent and Computer LawSouthfield

Personalised recommendations