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Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions

Author

Listed:
  • C. M. Yang

    (California Institute of Technology)

  • J. L. Beck

    (California Institute of Technology)

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.

Suggested Citation

  • C. M. Yang & J. L. Beck, 1998. "Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 211-227, April.
    • Handle: RePEc:spr:joptap:v:97:y:1998:i:1:d:10.1023_a:1022635419332
    DOI: 10.1023/A:1022635419332
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