Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Linearized ridge-path method for function minimization

  • 54 Accesses

  • 2 Citations

Abstract

A modification based on a linearization of a ridge-path optimization method is presented. The linearized ridge-path method is a nongradient, conjugate direction method which converges quadratically in half the number of search directions required for Powell's method of conjugate directions. The ridge-path method and its modification are compared with some basic algorithms, namely, univariate method, steepest descent method, Powell's conjugate direction method, conjugate gradient method, and variable-metric method. The assessment indicates that the ridge-path method, with modifications, could present a promising technique for optimization.

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

References

  1. 1.

    Courant, R.,Variational Methods for the Solution of Problems of Equilibrium Vibrations, Bulletin of the American Mathematical Society, Vol. 49, pp. 1–23, 1943.

  2. 2.

    Curry, H. B.,The Method of Steepest Descent for Nonlinear Minimization Problems, Quarterly of Applied Mathematics, Vol. 2, pp. 258–261, 1944.

  3. 3.

    Householder, A. S.,Principles of Numerical Analysis, McGraw-Hill Book Company, New York, New York, 1953.

  4. 4.

    Levenberg, K.,A Method for the Solution of Certain Nonlinear Problems in Least Squares, Quarterly of Applied Mathematics, Vol. 2, pp. 164–168, 1944.

  5. 5.

    Booth, A. D.,Numerical Methods, Butterworths, London, England, 1957.

  6. 6.

    Marquardt, D. W.,Solution of Nonlinear Chemical Engineering Models, Chemical Engineering Progress, Vol. 55, pp. 65–70, 1959.

  7. 7.

    Shah, B. V., Buehler, R. J., andKempthorne, O.,The Method of Parallel Tangents (PARTAN) for Finding an Optimum, Office of Naval Research, Report NR-042-207, 1961.

  8. 8.

    Fox, R. L.,Optimization Methods for Engineering Design, Addison-Wesley Publishing Company, Reading, Massachusetts, 1971.

  9. 9.

    Mangasarian, O. L.,Techniques of Optimization, Journal of Engineering for Industry, Vol. 94, pp. 365–372, 1972.

  10. 10.

    Walsh, G. R.,Methods of Optimization, John Wiley and Sons, New York, New York, 1975.

  11. 11.

    Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, pp. 163–168, 1963.

  12. 12.

    Davidson, W. C.,Variable Metric Method for Minimization, Atomic Energy Commission, Research and Development Report No. ANL-5990, 1959.

  13. 13.

    Fletcher, R., andReeves, C. M.,Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, pp. 149–154, 1964.

  14. 14.

    Hestenes, M. R., andStiefel, E.,Methods of Conjugate Gradients for Solving Linear Systems, Journal of Research of the National Bureau of Standards, Vol. B49, pp. 409–436, 1952.

  15. 15.

    Beckman, F. S.,The Solution of Linear Equations by the Conjugate Gradient Method, Mathematical Methods for Digital Computers, Vol. 1, Edited by A. Ralston and H. S. Wilf, John Wiley and Sons, New York, New York, 1960.

  16. 16.

    Rosenbrock, H. H.,Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal, Vol. 3, pp. 175–184, 1960.

  17. 17.

    Swann, W. H.,Report on the Development of a New Direct Search Method of Optimization, Imperial Chemical Industry Limited, Central Instrument Laboratory, Research Note No. 64/3, 1964.

  18. 18.

    Spendley, W., Hext, G. R., andHimsworth, F. R.,Sequential Applications for Simplex Designs in Optimization and Evaluating Operation, Technometrics, Vol. 4, pp. 441–461, 1962.

  19. 19.

    Nelder, J. A., andMead, R.,A Simplex Method for Function Minimization, Computer Journal, Vol. 7, pp. 308–313, 1965.

  20. 20.

    Powell, M. J. D.,An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives, Computer Journal, Vol. 7, pp. 155–162, 1964.

  21. 21.

    Friedman, M., andSavage, L. S.,Selected Techniques of Statistical Analysis, McGraw-Hill Book Company, New York, New York, 1947.

  22. 22.

    Spang, H. A.,A Review of Minimization Techniques for Nonlinear Functions, SIAM Review, Vol. 4, pp. 373–405, 1962.

  23. 23.

    Hooke, R., andJeeves, T. A.,Direct Search Solution of Numerical and Statistical Problems, Journal of the Association for Computing Machinery, Vol. 8, pp. 212–229, 1961.

  24. 24.

    Metwalli, S. M.,Optimum Design of Mechanical Isolation Systems for Vehicular Applications, State University of New York at Buffalo, PhD Dissertation, 1973.

  25. 25.

    Metwalli, S. M., andMayne, R. W.,New Optimization Techniques, ASME Paper No. 77-DET-169, 1977.

  26. 26.

    Gallagher, R. H., andZienkiewicz, O. C.,Optimum Structural Design, Theory and Applications, John Wiley and Sons, New York, New York, 1973.

  27. 27.

    Beveridge, G. S. G., andSchechter, R. S.,Optimization Theory and Practice, McGraw-Hill Book Company, New York, New York, 1970.

  28. 28.

    Phillips, D. A.,A Preliminary Investigation of Function Optimization by a Combination of Methods, Computer Journal, Vol. 17, pp. 75–79, 1971.

Download references

Author information

Additional information

This work was in partial fulfillment of the requirements for the MS degree of the first author at Cairo University, Cairo, Egypt. The authors would like to acknowledge the helpful and constructive suggestions of the reviewer.

Communicated by H. Y. Huang

Rights and permissions

Reprints and Permissions

About this article

Cite this article

TElzoughby, A.A., Metwalli, S.M. & Shawki, G.S.A. Linearized ridge-path method for function minimization. J Optim Theory Appl 30, 161–179 (1980). https://doi.org/10.1007/BF00934494

Download citation

Key Words

  • Optimization techniques
  • nonlinear programming
  • direct methods
  • numerical methods
  • conjugate directions
  • nongradient methods
  • ridge-path methods