Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions

Abstract

Two generalized trajectory methods are combined to provide a novel and powerful numerical procedure for systematically finding multiple local extrema of a multivariable objective function. This procedure can form part of a strategy for global optimization in which the greatest local maximum and least local minimum in the interior of a specified region are compared to the largest and smallest values of the objective function on the boundary of the region. The first trajectory method, a homotopy scheme, provides a globally convergent algorithm to find a stationary point of the objective function. The second trajectory method, a relaxation scheme, starts at one stationary point and systematically connects other stationary points in the specified region by a network of trjectories. It is noted that both generalized trajectory methods actually solve the stationarity conditions, and so they can also be used to find multiple roots of a set of nonlinear equations.

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References

  1. 1.

    Arora, J. S., Elwakeil, O. A., Chahande, A. I., and Hsieh, C. C., Global Optimization Methods for Engineering Applications: A Review, Structural Optimization, Vol. 9,Nos. 3–4, pp. 137–159, 1995.

    Google Scholar 

  2. 2.

    TÖrn, A., and Žilinskas, A., Global Optimization, Lecture Notes in Computer Science, Springer Verlag, Berlin, Germany, Vol. 350, 1989.

    Google Scholar 

  3. 3.

    Garcia, C. B., and Gould, F. J., Relations between Several Path-Following Algorithms and Local and Global Newton Methods, SIAM Review, Vol. 22,No. 3, pp. 263–274, 1980.

    Google Scholar 

  4. 4.

    Keller, H. B., Lectures on Numerical Methods in Bifurcation Problems, Springer Verlag, Heidelberg, Germany, 1987.

    Google Scholar 

  5. 5.

    Allgower, E., and Georg, K., Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations, SIAM Review, Vol. 22,No. 1, pp. 28–85, 1980.

    Google Scholar 

  6. 6.

    Watson, L. T., Billups, S. C., and Morgan, A. P., Algorithm 652—HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms, ACM Transactions on Mathematical Software, Vol. 13,No. 3, pp. 281–310, 1987.

    Google Scholar 

  7. 7.

    Desa, C., Irani, K. M., Ribbens, C. J., Watson, L. T., and Walker, H. F., Preconditioned Iterative Methods for Homotopy Curve Tracking, SIAM Journal on Scientific and Statistical Computing, Vol. 13,No. 1, pp. 30–46, 1992.

    Google Scholar 

  8. 8.

    Guillemin, V., and Pollack, A., Differential Topology, Prentice-Hall, Englewood Cliffs, New Jersey, 1974.

    Google Scholar 

  9. 9.

    Chow, S. N., Mallet-Paret, J., and Yorke, J. A., Finding Zeros of Maps: Homotopy Methods That Are Constructive with Probability One, Mathematics of Computation, Vol. 32, pp. 887–899, 1978.

    Google Scholar 

  10. 10.

    Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, Springer Verlag, New York, New York, 1995.

    Google Scholar 

  11. 11.

    Golubitsky, M., and Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, Vol. 1, Springer Verlag, New York, New York, 1985.

    Google Scholar 

  12. 12.

    Doedel, E., Keller, H. B., and Kernevez, J. P., Numerical Analysis and Control of Bifurcation Problems, I: Bifurcation in Finite Dimensions, International Journal of Bifurcation and Chaos, Vol. 1,No. 3, pp. 493–520, 1991.

    Google Scholar 

  13. 13.

    Kearfott, R. B., Some General Bifurcation Techniques, SIAM Journal on Scientific and Statistical Computing, Vol. 4,No. 1, pp. 52–68, 1983.

    Google Scholar 

  14. 14.

    Fujii, F., and Choong, K. K., Branch Switching in Bifurcation of Structures, ASCE Journal of Engineering Mechanics, Vol. 118,No. 8, pp. 1578–1596, 1992.

    Google Scholar 

  15. 15.

    Diener, I., On the Global Convergence of Path-Following Methods to Determine All Solutions to a System of Nonlinear Equations, Mathematical Programming, Vol. 39,No. 2, pp. 181–188, 1987.

    Google Scholar 

  16. 16.

    Diener, I., and Schaback, R., An Extended Continuous Newton Method, Journal of Optimization Theory and Applications, Vol. 67,No. 1, pp. 57–77, 1990.

    Google Scholar 

  17. 17.

    Seader, J. D., Kuno, M., Lin, W. J., Johnson, S. A., Unsworth, K., and Wiskin, J. W., Mapped Continuation Methods for Computing All Solutions to General Systems of Nonlinear Equations, Computers and Chemical Engineering, Vol. 14,No. 1, pp. 71–85, 1990.

    Google Scholar 

  18. 18.

    Beck, J. L., and Katafygiotis, L. S., Updating of a Model and Its Uncertainties Utilizing Dynamic Test Data, Proceedings of the 1st International Conference on Computational Stochastic Mechanics, Computational Mechanics Publications, Boston, Massachusetts, pp. 125–136, 1991.

  19. 19.

    Udwadia, F. E., Some Uniqueness Results Related to Soil and Building Structural Identification, SIAM Journal on Applied Mathematics, Vol. 45,No. 4, pp. 674–685, 1985.

    Google Scholar 

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Yang, C.M., Beck, J.L. Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions. Journal of Optimization Theory and Applications 97, 211–227 (1998). https://doi.org/10.1023/A:1022635419332

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  • Homotopy
  • relaxation
  • trajectory tracking
  • global optimization
  • roots
  • nonlinear equations