Journal of Optimization Theory and Applications

, Volume 97, Issue 1, pp 211–227

Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions

  • C. M. Yang
  • J. L. Beck

DOI: 10.1023/A:1022635419332

Cite this article as:
Yang, C.M. & Beck, J.L. Journal of Optimization Theory and Applications (1998) 97: 211. doi:10.1023/A:1022635419332


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.

Homotopy relaxation trajectory tracking global optimization roots nonlinear equations 

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • C. M. Yang
    • 1
  • J. L. Beck
    • 2
  1. 1.Division of Engineering and Applied SciencesCalifornia Institute of TechnologyPasadena
  2. 2.Division of Engineering and Applied SciencesCalifornia Institute of TechnologyPasadenaProfessor

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